# Supplementary mathematics/Printable version

Supplementary mathematics

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# Definition

بسم الله الرحمن الرحيم

Welcome to the Wikibook of SUPPLEMENTARY MATHEMATICS

This book is currently being designed for its introduction, and after the completion of the introductions, we will add the rest of the information to the others.

## Definition

This book is a guide for those interested in mathematics, which presents an advanced and complementary type of mathematics. In this book, we discuss advanced topics such as calculations, analysis, geometry, etc., and general topics such as the branches of mathematics. This book is different from basic math, basic math teaches basic concepts and teaches math in simple language. The concept of advanced mathematics means to present complex and advanced concepts, it means that extensive concepts are also included with them. This ebook will help you with advanced and extensive and important concepts of mathematics.

## Introduction

1. Introduction

## Mathematics analysis

1. Real analysis
2. Mixed analysis
3. Functional analysis
4. Harmonic analysis
5. Complex analysis
6. numerical analysis
7. Vector analysis

# Introduction

Mathematics is a subject in science that examines numbers, shapes, etc. Since mathematics is complex and important in life, it ranks first among all science subjects.

Mathematics is one of the best sciences in the world, followed by natural sciences. In this book, we discuss advanced and complex mathematical topics. This science is the best of all sciences and mathematics is very useful in physics, chemistry, engineering, astronomy, architecture, etc.

Mathematics is difficult from a distance, but if you get close to it, it is not difficult at all.

In this book, we also deal with related and important topics of mathematics. In this book, we discuss topics such as mathematics, calculus, geometry, and analysis, and examine important concepts, branches of mathematics, research fields, etc. This ebook has both a printed version and is a group collaboration.

Items:

1- Mathematics

2-calculus

3-Geometry

4- Discrete mathematics

5-probability statistics

6- Mathematical analysis

# Mathematics

Mathematics is the art of calculating numbers and also studies topics such as quantity (number theory), structure (algebra), space (geometry) and variation (mathematical analysis). In fact, there is no universal definition of mathematics that everyone agrees on.

Most mathematical activities involve discovering and proving the properties of abstract objects by pure reasoning. These objects are either abstractions of nature, such as natural numbers or lines, or – in modern mathematics – entities that have certain properties called axioms. An argument consists of a set of applications of some deductive rules to already known results, including previously proven theorems, axioms, and (if abstracted from nature) some basic properties that serve as the actual starting point of the theory under consideration. is considered are taken, the result of an argument is called a theorem.

Mathematics is widely used in science to model phenomena. This allows the derivation of quantitative predictions from empirical laws. For example, the motion of the planets can be accurately predicted using Newton's law of gravitation combined with mathematical calculations. The independence of mathematical truth from any experiment shows that the accuracy of such predictions depends only on the adequacy of the model to describe reality. Incorrect predictions indicate that the mathematical models need to be improved or changed, not that the mathematics in the models themselves are wrong. For example, the precession of Mercury's perihelion cannot be explained by Newton's law of gravitation, but it is precisely explained by Newton's law of gravitation. Einstein's General Relativity - This experimental confirmation of Einstein's theory shows that Newton's law of gravitation is only an approximation, although it is accurate in everyday applications.

Mathematics is essential in many fields including natural sciences, engineering, medicine, finance, computer science and social sciences. Some areas of mathematics, such as statistics and game theory, have been developed in close connection with their applications and are often grouped under applied mathematics. Other areas of mathematics develop independently of any application (and are therefore called pure mathematics), but practical applications are often discovered later. A good example is the problem of factoring integers, which dates back to the time of Euclid, but had no practical application before being used in the RSA cryptosystem (for computer network security).

Historically, the concept of proof and the mathematical precision associated with it first appeared in Greek mathematics, particularly in Euclid's Elements. From the beginning, mathematics was basically divided into geometry and calculus (manipulation of numbers and natural fractions) until in the 16th and 17th centuries, algebra and infinitesimal calculus were introduced as new disciplines. Since then, the interaction between mathematical innovations and scientific discoveries has led to the rapid growth of mathematics. At the end of the 19th century, the fundamental crisis of mathematics led to the systematization of the axiomatic method. This in turn caused a significant increase in the number of mathematical disciplines and their applied fields. An example of this classification is mathematics courses, which lists more than sixty areas of first level mathematics.

## History

The history of mathematics can be seen as a sequence of increasing abstractions. The first abstraction capability shared by many animals is probably the concept of number: the understanding that a set of two apples and a set of two oranges (for example) have in common, and that quantity is their number.

Prehistoric people could count both physical objects and abstract objects such as days, seasons, and years, as evidenced by woodcuts.

Evidence for more complex mathematics is not seen until 3000 BCE, when the Babylonians and Egyptians began using arithmetic, algebra, and geometry for calculations related to taxation and other economic concepts, and construction or astronomy. The oldest mathematical texts are from Mesopotamia and Egypt, dating back to 2000-1800 BC.Many early texts mention Pythagorean triads, so it seems that the Pythagorean theorem is one of the most important methods for inventing trigonometry.

This theorem has been proven many times by different geometric and algebraic methods, some of these proofs go back thousands of years. It is the oldest and most extensive mathematical development after elementary arithmetic and geometry. In historical documents, it was in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication and division) first appeared. The Babylonians also used a place-value device that implemented a base 60 number device, which is still used to measure angles and time.

With the beginning of the 6th century BC, the Greek mathematicians with the Pythagoreans began a systematic study of mathematics, with the aim of knowing more about mathematics itself, which was the beginning of Greek mathematics. Around 300 BC, Euclid introduced the method of thematic principles still used in mathematics, which included definitions, principles, theorems, and proofs. His reference book known as Euclid's Principles is widely regarded as the most successful and influential reference book of all time. The greatest ancient mathematicians are often considered to be Archimedes (287-212 BC) from Syracuse. He found formulas for calculating the area and volume of rotating objects and used Afna's method to calculate the area under a parabolic curve using the sum of an infinite series. In a way that is not dissimilar to modern calculus. Other notable achievements in Greek mathematics were conic sections (Apollonius of Perga, 3rd century BC), trigonometry (Hiparchus of Nicaea (2nd century BC)), and the beginning of algebra (Diophantus, 3rd century AD).

The Indo-Arabic number system and the rules for using its operations, which are used all over the world today, were developed in India during the first millennium AD and then transferred to the Western world through Islamic mathematics. Other developments related to Indian mathematics include the modern definition of sine and cosine and the early form of infinite series.

During the Golden Age of Islam, which took place in the 9th and 10th centuries AD, mathematics saw important innovations that were based on the mathematics of the Greeks. The most important achievements of Islamic mathematics was the development of algebra. Other important achievements of mathematics in the Islamic period were the progress in spherical trigonometry and the addition of decimals to the Arabic numerical system. Many mathematicians of this period were Persian-speaking, such as Khwarazmi, Khayyam and Sharafuddin Tosi.

During the early modern era, mathematics began to develop rapidly in Western Europe. The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonard Euler was the most important mathematician of the 18th century who added several theorems and discoveries to mathematics. Perhaps the most important mathematicians of the 19th century was the German mathematician Carl Friedrich Gauss, who provided many services to various branches of mathematics such as algebra, analysis, differential geometry, matrix theory, number theory, and statistics. In the early 20th century, Kurt Gödel revolutionized mathematics by publishing his incompleteness theorems. These theorems showed that any system of principles of compatibility includes unprovable propositions.

Since then, mathematics has been widely developed, and fruitful interactions between mathematics and science have been created that benefit both. Mathematical discoveries continue to this day. According to Mikhail Suriuk, published in the January 2006 Bulletin of the American Mathematical Society, "the number of articles and books in the Mathematical Review database since 1940 (MR's first year of operation) now stands at 1.9 million, an annual increase of over 75 Thousands of items will be added to this database.The vast majority of works in this ocean contain new mathematical theorems and their proofs.

## Areas of mathematics

Before the Renaissance, mathematics was divided into two main areas: arithmeticregarding the manipulation of numbers, and geometry, regarding the study of shapes. Some types of pseudoscience, such as numerology and astrology, were not then clearly distinguished from mathematics.

During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of the study and the manipulation of formulas. Calculus, consisting of the two subfields differential calculus and integral calculus, is the study of continuous functions, which model the typically nonlinear relationships between varying quantities, as represented by variables. This division into four main areasarithmetic, geometry, algebra, calculusendured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become a separate branch of mathematics until the seventeenth century.

At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than first-level areas. Some of these areas correspond to the older division, as is true regarding number theory (the modern name for higher arithmetic) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations.

### Number theory This is the Ulam spiral, which illustrates the distribution of prime numbers. The dark diagonal lines in the spiral hint at the hypothesized approximate independence between being prime and being a value of a quadratic polynomial, a conjecture now known as Hardy and Littlewood's Conjecture F.

Number theory began with the manipulation of numbers, that is, natural numbers $(\mathbb {N} ),$  and later expanded to integers $(\mathbb {Z} )$  and rational numbers $(\mathbb {Q} ).$  Number theory was once called arithmetic, but nowadays this term is mostly used for numerical calculations. Number theory dates back to ancient Babylon and probably China. Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler. The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss.

Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example is Fermat's Last Theorem. This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles, who used tools including scheme theory from algebraic geometry, category theory, and homological algebra. Another example is Goldbach's conjecture, which asserts that every even integer greater than 2 is the sum of two prime numbers. Stated in 1742 by Christian Goldbach, it remains unproven despite considerable effort.

Number theory includes several subareas, including analytic number theory, algebraic number theory, geometry of numbers (method oriented), diophantine equations, and transcendence theory (problem oriented).

### Geometry On the surface of a sphere, Euclidian geometry only applies as a local approximation. For larger scales the sum of the angles of a triangle is not equal to 180°.

Geometry is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines, angles and circles, which were developed mainly for the needs of surveying and architecture, but has since blossomed out into many other subfields.

A fundamental innovation was the ancient Greeks' introduction of the concept of proofs, which require that every assertion must be proved. For example, it is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results (theorems) and a few basic statements. The basic statements are not subject to proof because they are self-evident (postulates), or are part of the definition of the subject of study (axioms). This principle, foundational for all mathematics, was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements.

The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane (plane geometry) and the three-dimensional Euclidean space.

Euclidean geometry was developed without change of methods or scope until the 17th century, when René Descartes introduced what is now called Cartesian coordinates. This constituted a major change of paradigm: Instead of defining real numbers as lengths of line segments (see number line), it allowed the representation of points using their coordinates, which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry was split into two new subfields: synthetic geometry, which uses purely geometrical methods, and analytic geometry, which uses coordinates systemically.

Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions, the study of which led to differential geometry. They can also be defined as implicit equations, often polynomial equations (which spawned algebraic geometry). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions.

In the 19th century, mathematicians discovered non-Euclidean geometries, which do not follow the parallel postulate. By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics. This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not a mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space.

Today's subareas of geometry include:

• Projective geometry, introduced in the 16th century by Girard Desargues, extends Euclidean geometry by adding points at infinity at which parallel lines intersect. This simplifies many aspects of classical geometry by unifying the treatments for intersecting and parallel lines.
• Affine geometry, the study of properties relative to parallelism and independent from the concept of length.
• Differential geometry, the study of curves, surfaces, and their generalizations, which are defined using differentiable functions.
• Manifold theory, the study of shapes that are not necessarily embedded in a larger space.
• Riemannian geometry, the study of distance properties in curved spaces.
• Algebraic geometry, the study of curves, surfaces, and their generalizations, which are defined using polynomials.
• Topology, the study of properties that are kept under continuous deformations.
• Algebraic topology, the use in topology of algebraic methods, mainly homological algebra.
• Discrete geometry, the study of finite configurations in geometry.
• Convex geometry, the study of convex sets, which takes its importance from its applications in optimization.
• Complex geometry, the geometry obtained by replacing real numbers with complex numbers.

### Algebra

Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were the two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in the title of his main treatise.

Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas.

Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra), and polynomial equations in a single unknown, which were called algebraic equations (a ter m still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices, modular integers, and geometric transformations), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of a set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra, as established by the influence and works of Emmy Noether. (The latter term appears mainly in an educational context, in opposition to elementary algebra, which is concerned with the older way of manipulating formulas.)

Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include:

• group theory;
• field theory;
• vector spaces, whose study is essentially the same as linear algebra;
• ring theory;
• commutative algebra, which is the study of commutative rings, includes the study of polynomials, and is a foundational part of algebraic geometry;
• homological algebra;
• Lie algebra and Lie group theory;
• Boolean algebra, which is widely used for the study of the logical structure of computers.

The study of types of algebraic structures as mathematical objects is the purpose of universal algebra and category theory. The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces; this particular area of application is called algebraic topology.

### Calculus and analysis A Cauchy sequence consists of elements that become arbitrarily close to each other as the sequence progresses (from left to right).

Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz. It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts.

Analysis is further subdivided into real analysis, where variables represent real numbers, and complex analysis, where variables represent complex numbers. Analysis includes many subareas shared by other areas of mathematics which include:

• Multivariable calculus
• Functional analysis, where variables represent varying functions;
• Integration, measure theory and potential theory, all strongly related with probability theory on a continuum;
• Ordinary differential equations;
• Partial differential equations;
• Numerical analysis, mainly devoted to the computation on computers of solutions of ordinary and partial differential equations that arise in many applications.

### Discrete mathematics A diagram representing a two-state Markov chain. The states are represented by 'A' and 'E'. The numbers are the probability of flipping the state.

Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithmsespecially their implementation and computational complexity play a major role in discrete mathematics.

The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of the 20th century. The P versus NP problem, which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems.

Discrete mathematics includes:

• Combinatorics, the art of enumerating mathematical objects that satisfy some given constraints. Originally, these objects were elements

or subsets of a given set; this has been extended to various objects, which establishes a strong link between combinatorics and other parts of discrete mathematics. For example, discrete geometry includes counting configurations of geometric shapes

• Graph theory and hypergraphs
• Coding theory, including error correcting codes and a part of cryptography
• Matroid theory
• Discrete geometry
• Discrete probability distributions
• Game theory (although continuous games are also studied, most common games, such as chess and poker are discrete)
• Discrete optimization, including combinatorial optimization, integer programming, constraint programming

### Mathematical logic and set theory

The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic, although used for mathematical proofs, belonged to philosophy and was not specifically studied by mathematicians.

Before Cantor's study of infinite sets, mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration. Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument. This led to the controversy over Cantor's set theory.

In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour. Examples of such intuitive definitions are "a set is a collection of objects", "natural number is what is used for counting", "a point is a shape with a zero length in every direction", "a curve is a trace left by a moving point", etc.

This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory. Roughly speaking, each mathematical object is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic, the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism, as founded by David Hilbert around 1910.

The "nature" of the objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinionsometimes called "intuition"to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains the natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer, who promoted intuitionistic logic, which explicitly lacks the law of excluded middle.

These problems and debates led to a wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory, type theory, computability theory and computational complexity theory. Although these aspects of mathematical logic were introduced before the rise of computers, their use in compiler design, program certification, proof assistants and other aspects of computer science, contributed in turn to the expansion of these logical theories.

### Statistics and other decision sciences Whatever the form of a random population distribution (μ), the sampling mean (x̄) tends to a Gaussian distribution and its variance (σ) is given by the central limit theorem of probability theory.

The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory. Statisticians generate data with random sampling or randomized experiments. The design of a statistical sample or experiment determines the analytical methods that will be used. Analysis of data from observational studies is done using statistical models and the theory of infer ence, using model selection and estimation. The models and consequential predictions should then be tested against new data.

Statistical theory studies decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints. For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence. Because of its use of optimization, the mathematical theory of statistics overlaps with other decision sciences, such as operations research, control theory, and mathematical economics.

### Computational mathematics

Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis broadly includes the study of approximation and discretization with special focus on rounding errors. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic-matrix-and-graph theory. Other areas of computational mathematics include computer algebra and symbolic computation.

# Calculus

Calculus, which in the past was called calculus of infinitesimals, is a branch of mathematics. Just as geometry is the study of shapes and algebra is the generalization of arithmetic operations (the four main operations), arithmetic is the mathematical study of continuous changes.

Accounts have two branches: differential account and integral account. Differential calculus studies the rate of change and slope of the curves, while integral calculus deals with the accumulation of values and areas under the curves. These two branches are connected to each other by the fundamental theorem of calculus and use the fundamental concepts of convergence of sequences and infinite series to a well-defined limit.

Calculus of infinitesimals was independently developed in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Today, calculations have found wide applications in science, engineering and economics.

## History

### Antiquity

In the ancient period, some ideas led to integral calculus. But it does not seem that these ideas have led to a systematic and stable approach. Calculation of volume and area is one of the purposes of integral calculus, which can be found in the Moscow papyrus (13th Dynasty of Egypt, around 1820 BC); But the formulas were simple recipes without any indication of a specific method, so that some of these recipes lacked the main ingredients.

From the age of Greek mathematics, Eudoxus (c. 408-355 BC) used the method of Afna (who did something similar before discovering the concept of limit) to calculate areas and volumes, while Archimedes (ca. 212-287 BC) developed this idea further to invent a heuristic method that resembles the methods of integral calculus.

### Middle Ages

In the Middle East, Ibn Haytham (Latin: Alhazen) (965-1040 AD) derived a formula for the sum of fourth powers. He used the results of what we now call the integration of this function, such formulas for the sum of the square of integers and the fourth power also provided him with the possibility of calculating the volume of the parabola.

In the 14th century, Indian mathematicians presented an unstable method similar to differentiation that could be applied to some trigonometric functions.

In Europe, fundamental work took place in the form of Bonaventura Cavalieri's treatise. He was the one who claimed that volumes and areas should be written as sums of volumes and areas with infinitesimally small sections. These ideas were similar to the work of Archimedes in his treatise called "Method", but they believe that the mentioned treatise of Archimedes was lost in the 13th century and was rediscovered in the 20th century, so Cavalieri was not aware of its existence.

### Modern Era

The formal study of calculus brought together Cavalieri's method of infinitesimals and calculus of finite differences, which was developed in Europe at the same time. Pierre de Fermat claimed that the concept of "as equal as possible" (he coined the word "adequality" for this concept with the help of the Latin language) was inspired by Diophantus. This concept represented equality within an infinitesimal error term. The combination of these concepts was achieved by John Willis, Isaac Barrow and James Gregory, the latter two of whom proved the Second Fundamental Theorem of Arithmetic around 1670.

The rule of multiplication and chain rule, the concepts of derivatives of higher orders and Taylor's series, and analytical functions were used by Isaac Newton using a strange notation to solve problems in mathematics and physics. In his works, Newton restated his ideas in such a way as to correspond to the method of the time, as he replaced the calculations of infinitesimals with their geometrical equivalents. He used to solve problems such as the movement of the planets, the shape of the surface of a rotating fluid, the expansion of the globe at the poles (swelling at the poles), the movement of weight by sliding on a wheel, and many other problems that are in his work (the book Principia Mathematica written in 1687 AD) discussed, used the arithmetic method. In his other works, he used series expansions for functions, including fractional and non-exponential powers, so that it was clear that he understood the principle of Taylor's series. But he did not publish all these discoveries, and at that time, the use of the infinitesimal method was still in bad history and did not have a suitable aspect.

These ideas led to the calculus of real infinitesimals, organized by Gottfried Wilhelm Leibniz. Newton initially accused Leibniz of plagiarism. He is now credited as an independent inventor and contributor to accounts. His contributions were to provide a set of clear rules for working with infinitesimally small values, which provided the possibility of calculating derivatives of the second and higher orders, and provided the multiplication rule and the chain rule in differential and integral form. Contrary to Newton, Leibniz paid a lot of attention to formalization, in such a way that he often spent days determining the appropriate symbol for concepts.

Today, both Leibniz and Newton are credited for the invention and independent development of calculus. Newton was the first to use calculus in general physics, and Leibniz was the first to use much of the notation used in modern calculus. The basic insights provided by both Newton and Leibniz include: the laws of differentiation and integration, derivatives of the second order and higher, and the concept of approximation using polynomial series. By Newton's time, the fundamental theorem of calculus was known.

Since Leibniz and Newton, many mathematicians have contributed to the continuous development of calculus. One of the first and most complete works on both the calculus of infinitesimals and the integral calculus was written in 1748 by Maria Gaetna Agnesi.

### Application

The use of infinitesimal calculus for physics and astronomy problems was contemporary with the birth of science. Throughout the 18th century, these applications multiplied, until Laplace and Lagrange brought a wide range of study of forces into the realm of analysis. We owe the introduction of potential theory to dynamics to Lagrange (1773), although the name "potential function" and the basic memories of the subject are due to Greene (1827, published in 1828). The name "potential" is due to Gauss (1840) and the distinction between potential and potential function to Clausius. With its development, the names of Lejon Dirichlet, Riemann, von Neumann, Heine, Kronecker, Lipschitz, Christoffel, Kirchhoff, Beltrami and many prominent physicists of the centuryi is

In this article, it is not possible to go into various other applications of analysis for physical problems. including Euler's research on vibrating chords. Sophie Germain on elastic membranes; Poisson, Lamé, Saint-Venant and Klebsch on the elasticity of three-dimensional bodies. February in heat release. Fresnel in light; Maxwell, Helmholtz, and Hertz on electricity. Hansen, Hill and Gilden on Astronomy. Maxwell on spherical harmonics. Lord Rayleigh on acoustics. and the contributions of Lejon Dirichlet, Weber, Kirchhoff, F. The labors of Helmholtz deserve special mention, as he contributed to the theories of dynamics, electricity, etc., and applied his great analytical power to the fundamental principles of mechanics, as well as to pure mathematics.

Furthermore, infinitesimal calculus entered the social sciences beginning with neoclassical economics. Today, it is a valuable tool in the mainstream economy.

## The basis and importance of accounts

### base

In calculus, "fundamentals" refers to the detailed development of the subject from axioms and definitions. In early calculations, the use of infinitesimally small values was thought to be imprecise and was strongly criticized by a number of authors, notably Michel Rolle and Bishop Berkeley. In 1734, Berkeley described infinities as the ghosts of vanishing quantities in his book The Analyst. Establishing a precise foundation for calculus preoccupied mathematicians throughout the century since Newton and Leibniz, and remains somewhat of an active area of research today.

Several mathematicians, including McLaren, tried to prove the use of the infinitesimally small, but it was not until 150 years later, due to the work of Cauchy and Weierstrass, that a way was finally found to avoid mere "notions" of the infinitesimally small. . The foundations of differential and integral calculus were laid. In Cauchy's Cours d'Analyse, we find a range of fundamental approaches, including the definition of continuity in terms of infinities, and the (somewhat imprecise) prototype of the (ε, δ)-limit definition in the definition of differentiation. In his work, Weierstrass formalized the concept of limit and eliminated the infinity of smalls (although his definition can actually confirm the infinity of zero-squared smalls). Following the work of Weierstrass, it eventually became common to base calculus on limits rather than infinitesimals, although this is still sometimes called "infinitesimal calculus". Bernhard Riemann used these ideas to give a precise definition of the integral. It was also during this period that with the development of complex analysis, the ideas of differential and integral calculus were extended to the complex level.

In modern mathematics, the basics of differential and integral calculus are included in the field of real analysis, which includes definitions and complete proofs of theorems of differential and integral calculus. Access to calculus has also been greatly expanded. Henri Lebego developed measure theory based on earlier developments by Emil Burrell and used it to define the integrals of all but the most pathological functions. Loren Schwartz introduced distributions that can be used to take the derivative of any function.

Limits are not the only exact approach to the foundation of calculus. Another way is to use Abraham Robinson's non-standard analysis. Robinson's approach, developed in the 1960s, uses technical machinery from mathematical logic to augment the real number system with infinitesimally small numbers, as in the original Newton-Leibniz concept. The resulting numbers are called metareal numbers and can be used to give a Leibnizian extension like the usual rules of arithmetic. There is also smooth infinitesimal analysis, which differs from nonstandard analysis in that it requires the neglect of infinitesimals of higher powers during the derivation.

### Importance

While many of the ideas of calculus had already been developed in Greece, China, India, Iraq, Iran, and Japan, the use of calculus began in the 17th century, when Isaac Newton and Gottfried Wilhelm Leibniz built on their work. were, it started in Europe. Earlier mathematicians introduced its basic principles. The development of calculus is based on the basic concepts of instantaneous motion and area under curves.

Applications of differential calculus include calculations related to speed and acceleration, curve slope and optimization. Applications of integral calculus include area, volume, arc length, center of mass, work and pressure calculations. More advanced programs include power series and Fourier series.

Calculus is also used to gain a more precise understanding of the nature of space, time, and motion. For centuries, mathematicians and philosophers have struggled with paradoxes involving division by zero, or the infinite sum of numbers. These questions are raised in the study of motion and area. The ancient Greek philosopher Zeno of Elea gave several famous examples of these paradoxes. Calculus provides tools, especially limits and infinite series, that resolve paradoxes.

## Principles

### Limits and infinitesimals

Calculus is often developed by working on very small values. Historically, the first method was done with the help of infinitesimals. These are objects that can be treated like real numbers, but are "infinitely small" in some respects. For example, an infinitesimally small number may be greater than zero, but less than any number in the sequence $1,{\frac {1}{2}},{\frac {1}{3}},\cdots$  and therefore it is smaller than any positive real number. From this point of view, arithmetic is a collection of infinitesimal manipulation techniques. The symbols $dx$  and $dy$  were considered to represent infinitesimals and the derivative, i.e. $dy/dx$ , was simply the ratio of the two.

The infinitesimals approach was abandoned in the 19th century because it was difficult to make the concept of infinitesimals precise. However, this concept was revived in the 20th century with the introduction of the concept of non-standard analysis and analysis of smooth infinitesimals, which provided a solid foundation for the manipulation of infinitesimals.

At the end of the 19th century, infinitesimals were replaced in scientific circles by the epsilon and delta approach to define the limit. It describes the range of values of a function at a particular input in terms of its values at adjacent inputs. This tool captures the small-scale behavior in the context of a real number machine. In this approach, calculus can be considered a collection of techniques for manipulating certain limits. The infinitesimals were replaced by very small numbers, and the infinitesimal behavior of a function is obtained by its limiting behavior for smaller and smaller numerical values. Limits were thought to provide a solid foundation for calculations, and for this reason, this approach became the standard approach in the 20th century.

### Differential Calculus tangent line at ( x , f ( x ) ) {\displaystyle (x,f(x))}  . The derivative f ′ ( x ) {\displaystyle f'(x)}   of a curve at a point is equal to the slope of the tangent line (the opposite side divided by the adjacent side in the corresponding right triangle) at that point.

Calculus studies the definition, properties and applications of the derivative of a function. The process of finding the derivative is called "differentiation". If we consider a function and a point in its domain, the derivative of that point is a method that includes the small-scale behavior of a function near that point. By finding the derivative of a function at any point of its domain, it is possible to generate a new function called "derivative function" or simply "derivative" of the main function. In formal language, the derivative is a linear operator that takes one function as input and produces another function as output. The latter description is more abstract than many of the processes studied in elementary algebra, where the input and output of the function were simply numbers. For example, if a doubling function were considered, an input of three would produce an output of six, and if a squaring function were considered, an input of three would produce an output of nine. While derivation takes the entire function of the square generator as input, that is, all the information related to what numerical output of each numerical input of that function goes, and based on that information, it builds another function, which is the doubling function. Is.

In more explicit language, the "doubling function" can be represented as $g(x)=2x$  and the "squaring function" as $f(x)=x^{2}$ . Now the "derivative" takes the function $f(x)$  defined by the expression "$x^{2}$ " as input, and from that the function $g(x)=2x$ .

The most common symbol for a derivative is a sign similar to an apostrophe, which is called prime (or prim in Persian). Therefore, the derivative of a function like $f$  is written as $f'$  and it is called "f prime". For example, if $f(x)=x^{2}$  is the squaring function, then $f'(x)=2x$  is its derivative (the same doubling function as in discussed above). This notation is known as Lagrange notation.

## Application

Calculus is used in every branch of physical science, actuarial science, computer science, statistics, engineering, economics, commerce, medicine, demography and in other fields wherever a problem can be mathematically modeled and Its optimal solution is found. desired. This allows one to go from (non-constant) rate of change to total change or vice versa, and many times in studying a problem we recognize one and try to find the other. Calculus can be used in conjunction with other mathematical disciplines. For example, it can be used with linear algebra to find the "best-fit" linear approximation for a set of points in a domain. Or it can be used in probability theory to determine the expected value of a continuous random variable according to the probability density function. In analytical geometry, to study graphs of functions, calculus is used to find high and low points (maximum and minimum), slope, concavity, and turning points. Calculus is also used to find approximate solutions of equations. to be In practice, this is the standard method for solving differential equations and finding roots in most applications. For example, there are methods such as Newton's method, fixed point iteration and linear approximation. For example, spacecraft use the modified Euler method to approximate curved trajectories in zero-gravity environments.

Physics makes special use of calculus. All concepts in classical mechanics and electromagnetism are related through calculus. The mass of an object of known density, the moment of inertia of the object, and the potential energy due to gravitational and electromagnetic forces can be found using calculus. An example of the use of calculus in mechanics is Newton's second law of motion, which states that the derivative of an object's momentum with respect to time is equal to the net force. On the other hand, Newton's second law can be expressed by saying that the net force is equal to It is expressed by the mass of the object times its acceleration, which is the time derivative of the velocity and therefore the second time derivative of the position. Starting from knowing how an object accelerates, we use calculus to derive its path.

Maxwell's electromagnetic theory and Einstein's theory of general relativity are also expressed in the language of differential calculus. Chemistry also uses calculus in determining reaction rates and in the study of radioactive decay. In biology, population dynamics starts with reproduction and mortality rates to model population changes.

Green's Theorem, which expresses the relationship between a line integral around a simple closed curve C and a double integral over the region of the plane D bounded by C, is applied to an instrument called a planometer, which is used to calculate the area of a flat. will be Area on a painting For example, it can be used to calculate the amount of area occupied by an irregularly shaped flower bed or swimming pool when designing the layout of a piece.

In the medical realm, calculus can be used to find the optimal bifurcation angle of a blood vessel in order to maximize flow. Calculus can be used to find out how fast a drug is eliminated from the body or how fast a cancerous tumor grows.

In economics, calculus enables the determination of maximum profit by providing a way to easily calculate marginal cost and marginal revenue.

## types

Over the years, many reformulations of calculus have been explored for various purposes.

### Non-standard account

Imprecise calculations with small infinitesimals were largely replaced by the exact definition of the (ε, δ) limit starting in the 1870s. Meanwhile, calculations continued with the infinitesimally small, often leading to correct results. This led Abraham Robinson to investigate whether it was possible to create a number system with infinitesimally small quantities where the theorems of calculus still hold. In 1960, relying on the works of Edwin Hoyt and Jerzy Losch, he succeeded in developing non-standard analysis. The theory of nonstandard analysis is rich enough to be applicable to many branches of mathematics. As such, books and articles devoted solely to traditional calculus theorems are often titled non-standard calculus.

### Smooth infinitesimal analysis

This is another reformulation of calculus in terms of the infinitesimal. Based on the ideas of FW Lawvere and using the methods of category theory, he considers all functions to be continuous and unable to express discrete entities. One of the aspects of this formulation is that there is no law of the eliminated middle in this formulation.

### constructive analysis

Constructive mathematics is a branch of mathematics that insists that proving the existence of a number, function, or other mathematical object must provide a structure of the object. In this way, constructive mathematics also rejects the law of the omitted middle. Reformulating calculus in a constructive framework is generally part of the subject of constructive analysis.

# Geometry

Geometry is a branch of mathematics that deals with the shapes of all physical objects, spatial and volume relationships between different objects, spatial features around and calculating the volume and area of shapes. This science is one of the oldest branches of the great sciences of mathematics, which emerged in response to practical problems such as mapping and had an important application in life. The name geometry is derived from the Greek and Latin words meaning "earth measurement". Finally, it became clear that geometry should not be limited to the study of flat surfaces (plane geometry) and rigid three-dimensional objects (solid geometry) and found that even the most abstract thoughts and images can be represented and developed in geometric forms.

# Discrete mathematics

Discrete mathematics is a part of mathematics dedicated to the study of discrete objects (discrete means distinct or unrelated elements); In other words, in response to the question of what is discrete mathematics, we can say that whenever objects are counted, relationships between finite (or countable) sets are studied, and in general, discrete mathematics is used in processes that involve a limited number of steps. Discrete mathematics study areas include counting, transformation (permutation), composition, graph theory, number theory, sets and relations, function and recursive relation.

From the time of the English scientist Isaac Newton until the late Renaissance period, almost 80% of all emphasis on the structures of branches of mathematical sciences such as applied mathematics and pure mathematics has been on continuously variable processes modeled by the mathematical continuum and derived from methods of calculus. Differential and integral and limits from branches of geometry such as: analytical geometry and spatial geometry) and statistics and probability are used. In contrast, discrete mathematics is mainly concerned with finite sets of discrete objects such as numbers and skeletal figures (such as graphs). With the growth of digital devices, especially computers, discrete mathematics has become more important.

Discrete structures can be counted, arranged, placed in collections, analysis can be done with mathematical logic, tabulation of numbers and calculations can be done and compared with each other. Although discrete mathematics is a broad and diverse discipline, it also has specific rules for mathematical, logical, physical, and geometrical formulas that exist in many subjects. The concept of independent events and resulting rules, sums, and PIEs is shared between compositions, set theory, and probability. In addition, De Morgan's laws are applicable to many areas of discrete mathematics.

Often, what makes discrete math problems interesting and challenging are the constraints placed on them. Although the discipline of discrete mathematics has many elegant formulas for applications, it is rare that a practical problem corresponds perfectly to a particular formula. Part of the fun of exploring discrete mathematics is learning about many different approaches to solving a problem, and then being able to creatively apply different strategies to a solution.

# Statistics and Probability

Statistics, in the popular sense of the term, deals with the group study of a population with the help of mathematics . In descriptive statistics , we simply describe a sample using quantities such as the mean, the median, the standard deviation, the proportion, the correlation, etc. This is often the technique that is used in censuses.

In a broader sense, statistical theory is used in research for inferential purposes. The goal of statistical inference is to draw the portrait of a given population, from the more or less blurred image formed using a sample from this population.

In another order of ideas, there is also “mathematical” statistics where the challenge is to find judicious (unbiased and efficient) estimators. The analysis of the mathematical properties of these estimators is at the heart of the work of the mathematician specializing in statistics.

The terms inductive statistics , appraisal statistics and inferential statistics ( inferential statistics ) are mostly used synonymously , which characterize the part of the statistics that is complementary to the descriptive statistics . Together with the theory of probability, mathematical statistics form the branch of mathematics known as stochastics .The mathematical basis of mathematical statistics is the theory of probability.

Probability theory in mathematics is the study of phenomena characterized by chance and uncertainty. Along with statistics , it forms the two sciences of chance which are an integral part of mathematics. The beginnings of the study of probabilities correspond to the first observations of chance in games or in climatic phenomena , for example.

Bell curve, histogram and dice.

Although the calculation of probabilities on questions related to chance has existed for a long time, the mathematical formalization is only recent. It dates from the beginning of the 20th century with Kolmogorov  's axiomatics . Objects such as events , probability measures , probability spaces or random variables are central in the theory. They make it possible to abstractly translate behaviors or measured quantities that can be assumed to be random. Depending on the number of possible values ​​for the random phenomenon studied, probability theory is said to be discrete or continuous.. In the discrete case, that is to say for at most a countable number of possible states, the theory of probabilities approaches the theory of enumeration  ; whereas in the continuous case, the theory of integration and the theory of measure provide the necessary tools.

Probabilistic objects and results are a necessary support for statistics , this is the case for example of Bayes' theorem , the evaluation of quantiles or the central limit theorem and the normal law . This modeling of chance also makes it possible to resolve several probabilistic paradoxes .

Whether discrete or continuous, stochastic calculus is the study of random phenomena that depend on time. The notion of stochastic integral and stochastic differential equation are part of this branch of probability theory. These random processes make it possible to make links with several more applied fields such as financial mathematics , statistical mechanics , image processing , etc.

# Mathematics analysis

Analysis is a branch of mathematics . Analysis has existed as an independent branch of mathematics since Leonhard Euler (18th century). Since then it has been the mathematics of natural and engineering sciences .

Its basics were developed independently in the 17th century by Gottfried Wilhelm Leibniz and Isaac Newton as infinitesimal calculus . Calculus is the mathematical study of continuous change, just as geometry is the study of form and algebra is the study of the generalization of arithmetic operations.

Central concepts of analysis are those of the limit value , the sequence , the series and, in particular, the concept of the function . The investigation of real and complex functions with regard to continuity , differentiability and integrability is one of the main subjects of analysis. The two bodies are fundamental to the entire analysis (the field of real numbers ) and(the field of complex numbers ) together with their geometric, arithmetic , algebraic and topological properties.

# Philosophy of mathematics

The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's lives. The logical and structural nature of mathematics makes this branch of philosophy broad and unique.

1. the mode of being of the mathematical objects: do they exist "really" and independently of a concrete use, and if so, in what sense? What does it even mean to refer to a mathematical object? What is the character of mathematical propositions? What are the relationships between logic and mathematics? – These are ontological questions .
2. the origin of mathematical knowledge : what is the source and essence of mathematical truth ? What are the conditions of mathematical science? What are your basic research methods? What role does human nature play in this? – These are epistemological questions .
3. the relationship between mathematics and reality : What is the relationship between the abstract world of mathematics and the material universe? Is mathematics anchored in experience , and if so, how? How is it that mathematics “fits the objects of reality so perfectly” ( Albert Einstein ) ? In what way do concepts like number , point , infinity gain their meaning beyond the inner-mathematical realm?

The starting point is almost always the view that mathematical propositions are apodictically certain, timeless and exact and that their correctness depends neither on empirical results nor on personal opinions. The task is to determine the conditions for the possibility of such knowledge, as well as to question this starting point.

# History of mathematics

The history of mathematics deals with the origin of discoveries in mathematics as well as mathematical methods and notation of the past. Written evidence of new developments in mathematics prior to the modern era and subsequent spread of knowledge around the world have been identified in only a few places. From about 3000–5000 BC, the Mesopotamian states of Sumer, Akkad, and Assyria, followed by ancient Egypt and the Syrian states of Abla and Elam began to use arithmetic, algebra and geometry for the purposes of taxation and trade. They were also used in architecture, drawing, and engraving, as well as in patterns found in nature, such as astronomy, time recording, and calendar editing.

The International Mathematical Olympiad (IMO) is a test of the Mathematical Olympiad for pre-university students.

Content ranges from very difficult problems in algebra and pre-calculus to problems in branches of mathematics not typically covered in middle school or high school and often not at the college level, such as pictorial and complex geometry, functional equations, combinations, and ... is somewhat well-contextualized. Number theory, which requires extensive knowledge of theorems, is well placed in this context. Calculus, although allowed in the solutions, is never required because the principle exists that anyone with a basic understanding of mathematics should understand the problems, even if the solutions require much more knowledge. to be Proponents of this principle claim that this allows for greater globalization and creates an incentive to find solutions to seemingly simple but enticing problems that nevertheless require a certain level of ingenuity, often a great deal of ingenuity, to solve. Getting all points for a given problem IMO.

The selection process varies by country, but often involves a series of tests that admit fewer students in each progressing test. Prizes will be awarded to approximately 50% of the top scoring participants. Teams are not officially recognized - all points are awarded to individual participants only, but team scores are unofficially compared to individual scores. Participants must be under 20 years of age and must not be in No higher institution to register. According to these conditions, a person can participate in IMO any number of times.

# Applied mathematics

Applied mathematics is a science that is used to apply mathematical methods in various fields such as physics, engineering, medicine, biology, finance, business, computer science and industry. Therefore, applied mathematics is a combination of mathematical sciences and specialized sciences. The term "applied mathematics" also describes a professional specialty in which mathematicians work on practical and scientific problems by formulating, studying, and thinking about mathematical models.

In the past, practical applications have also motivated the development and progress of mathematical theories, which then led to the invention of the science of pure mathematics, where abstract concepts are studied for their own sake and have exceptional methods. Therefore, the activity of applied mathematics is closely related to research in pure mathematics. Applied mathematics has concepts that are used in specialized fields. This topic has even been used in mathematics itself, for example, physical objects in nature. can be mentioned in geometry, slope length in differential and integral calculus, etc. It can be said that applied mathematics by discovering in physics, theories, etc It can be used.

# Pure mathematics

Pure mathematics is the study of mathematical concepts independent of any application outside of mathematics. These kinds of concepts may originate in real-world concerns, and the results obtained may later be highly useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the attraction is attributed to the intellectual challenge and the aesthetic beauty of working to the logical consequences of underlying principles.

While pure mathematics as an activity has existed at least since ancient Greece and has made effective progress, of course, these concepts can be seen in ancient Iran, ancient Egypt, ancient Babylon,And even in the golden period of Islam, etc. This concept was carried out around 1900, after the introduction of theories with counterintuitive features (such as non-Euclidean geometries and the theory of infinite contour sets) and complex concepts of calculus, and the theory was explained. The discovery of apparent paradoxes (such as continuous functions that are nowhere differentiable and Russell's paradox) was also discussed as an example in the scientific field. It introduced the need to renew the concept of mathematical precision and rewrite all mathematics based on it, with the systematic use of axiomatic methods. This led many mathematicians to focus on complete mathematics for its own sake, i.e. pure mathematics.

It follows that, presently, the distinction between pure and applied mathematics is more a philosophical point of view or a mathematician's preference rather than a rigid subdivision of mathematics. In particular, it is not uncommon that some members of a department of applied mathematics describe themselves as pure mathematicians.

# Computational mathematics

Computational mathematics includes mathematical research in branches of science where calculation plays an important and key role. Calculation means algorithms, numerical methods and symbolic methods. Calculation is leading in research. Computational mathematics emerged in the 1950s as a different branch of applied mathematics.

Today, it is necessary to use a computer to perform calculations to solve various scientific problems. Computational mathematics includes or relates to:

• Computing science, or scientific computing or computing engineering or engineering and computing science
• Solving mathematical problems with computer simulation, unlike applied mathematical analytical methods
• Numerical methods used in scientific calculations such as numerical linear algebra, numerical solution of partial differential equations
• Stochastic methods such as Monte Carlo methods and other representations of uncertainty in scientific computing, for example random finite elements
• Mathematics of scientific calculations, which from a mathematical point of view includes mathematical proofs, such as numerical analysis and the theory of numerical methods, and from the point of view of theoretical computer science, including the theory of calculations and computational complexity
• Symbolic calculations and computer algebra devices
• Research with the help of computer in various branches of mathematics such as logic (subsidized proof of proof), discrete mathematics, etc.
• Computational linguistics, the use of mathematical and computer techniques in natural languages
• Computational algebraic geometry
• Computational group theory
• Computational geometry
• Computational number theory
• Computational topology
• Computational statistics
• Algorithmic theory of information
• Algorithmic theory of games

# Branches of mathematics

Mathematics covers various types and depth of subjects throughout history, and only by sorting and categorizing all these subjects in mathematical branches can they be understood and collected in one place. Several models emerged to categorize these issues and although there are commonalities between these models, each one is different from the others due to their purpose.

Traditionally, mathematics is divided into pure (the study of mathematics for its inherent beauty) and applied (the study of mathematics for its application in real-world problems). But this general division was not always clear and many subjects were first founded by pure mathematics only to find its applications later. Major divisions such as discrete mathematics , computational mathematics , etc. have emerged recently.

An ideal taxonomy would allow new branches to be added to previous knowledge, making surprising improvements and being able to accommodate unexpected connections between branches to previous classifications. For example, Langlands' program found unexpected connections between previously considered unrelated fields, such as connections between Galva groups , Riemannian procedures , and number theory.

# Integral of Fourier

In differential and integral calculus, mathematical analysis and mathematical expansions, the Fourier integral or the Fourier integral operator has become an important element in the theory of numerical differential and integral equations. The class and function of the Fourier integral is applied in mathematics and as a branch of mathematics. The class of Fourier integral operators is as a differential operator, as well as classical integral operators as special cases.

# Volume integral

The three-dimensional integral, also known as the three-dimensional definite integral or volume integral, is an extension of the ordinary integral to three-dimensional space. It is often preferred to solve problems in three-dimensional space in dimensional spatial geometry, such as finding the center of mass, moments of inertia or the volume of a solid region under a curve and the exact volume of geometric volumes.

# Surface integral

In mathematics, a double integral or a surface integral is a type of dual-use 2D-3D integral that is used to add a group or data of values and intervals associated with points on a 2D and 3D surface. Calculating the surface integral is similar to calculating the surface area using the double integral except for the function inside the integral.

3D and 2D Cartesian, Cylindrical and Spherical coordinate fields in scalar continuous fields, algebraic, rational and numerical scalar value function and in 3D and 2D Heli fields, vector value function returns the vectors in coordinate or numerical form based on their dimensions. Like linear integrals, surface integrals are of two types:

1-Surface integral in scalar function (numerical function)

2-surface integral in vector function (coordinate function)

Surface integral or two-dimensional scalar function is also a simple generalization of double integral, while surface integral of vector functions plays an important role in the basic theorem of calculus.

# Trigonometry

Trigonometry is one of the most important and best arithmetic calculations. Of course, trigonometry has six components to calculate. Read the definition of trigonometry well.

### What is trigonometry?

Trigonometry is one of the branches of arithmetic and mathematics that examines the relationship between lengths and angles of triangles and makes a formulation for the relationship between angles and side lengths for triangles. In short, Shore said that scientific trigonometry is for Triangle is used to calculate the length and sides of a triangle. The word trigonometry comes from the Greek word (trigon, metron) which means triangle and size, respectively, and in the compound word, it means the measurement of a triangle.This concept was first given by the Greek mathematician Hipparchus. Trigonometry can only be calculated for right triangles and different triangles are not used.Trigonometry is divided into two hypothetical branches, spherical trigonometry (space) and circular trigonometry (plane).Angles in trigonometry are calculated in radians. Trigonometry, in general, is about trigonometric formulas, trigonometric ratios, and functions, right triangles, etc.

# Fourier series

The Fourier series is a periodic expansion for functions such as f(x) in terms of the sum of the infinity of the sine and cosine functions, and the expansion is exponential. is also used in the Fourier series. The study of the Fourier series is one of the calculus of calculus and is known as the analysis of harmonics. Of course, this topic can be an arbitrary tonic function in a trigonometric set. and integral to match.

In particular, since the superposition principle holds for solutions of a linear homogeneous ordinary differential equation, if such an equation can be solved for a single sinusoid, the solution of an arbitrary function is immediately available by expressing the original function in Fourier form. Is. Connect the series and then solve for each sinusoidal component. In some special cases where the Fourier series can be summed in closed form, this technique can even yield analytical solutions.

Any set of functions that form a complete orthogonal system has a generalized Fourier series similar to the Fourier series. For example, using the orthogonality of the roots of a Bessel function of the first kind yields a so-called Fourier-Bessel series.

## Fourier series for periodic functions

For example, expressions such as sine, cosine and exponential eikx can be used to define the Fourier series. It provided square waves (1 or 0 or -1) which are good examples, along with delta functions in the derivative. In order to present a spike and a step function, and a sloping surface, etc., Bayez looked at how their expansion should be found theoretically and mathematically. You should start with sin x first. This Fourier series theory has a period of 2π of sin(x + 2π) = sin x. It is an odd function because sin(-x) = - sin x, and vanishes at x = 0 and x = π. Every sin nx function has these three properties, and Fourier looked at infinite combinations of sines

$S(x)=b_{1}\sin x+b_{2}\sin 2x+b_{3}\sin 3x+...=\sum _{k=1}^{N}b_{n}\sin nx$

If we assume that the numbers in the Fourier series decrease quickly enough, the set S(x) will have three characteristics. In this hypothesis, the importance of the decay rate and code is predicted.

Periodic S(x + 2π) = S(x) Odd S(−x) = −S(x) S(0) = S(π) = 0

200 years ago, Joseph Fourier, a French mathematician, expanded the Fourier series with an interesting suggestion. Joseph Fourier realized that the series of the function S(x) with those properties can be written as an infinite series. He expressed sine and cosine. This idea started the huge and important development and development of the Fourier series. The first step for us and you in the Fourier series is to calculate the number bk that multiplies sin kx or cos kx from S(x).

If we assume that the expansion of $S(x)=\sum b_{n}\sin(nx)$ , both sides can be multiplied by sin kx. If it is integrated from 0 to π, it is integral:

$\int _{0}^{\pi }S(x)=\sin(kx)dx=\int _{0}^{\pi }b_{1}\sin(x)\sin(kx)dx+\int _{0}^{\pi }b_{2}\sin(2x)\sin(kx)dx+...+\int _{0}^{\pi }b_{k}\sin(kx)\sin(kx)dx$

On the right, all integrals are zero except for the highlighted integral with n = k.This "perpendicular" feature will dominate the entire chapter. They make sinuses 90◦ angles in function space, when their inner products are integral from 0 to π:

$\int _{0}^{\pi }\sin(nx)\sin(kx)dx$

# Taylor series

In mathematics, the Naylor series is an expression of a function, for example, f, in which all the derivatives of the order of magnitude of the order exist.

Σ ∞n = 0 f (n) (a) (z − a)n/n!

In this expression, sigma or Σ represents the sum of each element in the series, where n varies from zero (0) to infinity or (∞), f (n) represents the nth derivative of the function f and the expression n! The factorial function in the series is represented by the standard form. This series is named after Brooke Taylor, a mathematician and scientist from England.

# power series

A power series is a type of mathematical series that is also an infinite series. The back power series shows an infinite function that either converges or diverges. The application of the power series for identity or single-sentence patterns is to be able to calculate them as an orderly expansion and their convergence.

The power series of a single variable converges at the radius of convergence, which means that within the extent of this radius or region of convergence, all values of the variable less than the radius tend to converge to a point.

# Laplace's equation

In mathematical calculations and analysis, Laplace's equation or Laplace's finite differential equation is a second-order partial differential equation that is denoted by the divergence symbol ▽ and is used in mathematics, physics, vantage point calculus, geometry, engineering, etc. This is a useful approach and function to determine an energy or electrical potential in free space or area, which is used in energy and high degrees compared to its structure, and for mathematics for sine and cosine signal waves, etc. in spherical coordinates. and cylindrical and Cartesian are used. Laplace's equation was derived to simplify calculations in physics and is named after the physicist Pierre Simon Laplace.

# Laplace transform

In mathematics and calculus, the Laplace transform, named after its French discoverer, Pierre-Simon Laplace, is a transformation for calculus that transforms a function of a real variable (usually in the time domain) into a function of a The complex variable (in the complex frequency domain, also known as s-domain or s-plane) transforms. The Laplace transform has one of its many applications in science and engineering because it is a tool for solving Doing differential equations in calculus is differential and integral. Also, Laplace transformation can convert all ordinary differential equations into algebraic equations and convolution into multiplication. For suitable functions f, the Laplace transform is integral.

# Volume element

In mathematics and calculus and geometry, a volume element generally provides a means to integrate a function according to its position in the volume of different coordinate systems such as spherical coordinates and cylindrical coordinates. Therefore, a volume element is an expression of the form:

$\mathrm {d} V=\rho (u_{1},u_{2},u_{3})\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}\,\mathrm {d} u_{3}$

where the $u_{i}$  are the coordinates, so that the volume of any set $B$  can be computed by:$\operatorname {Volume} (B)=\int _{B}\rho (u_{1},u_{2},u_{3})\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}\,\mathrm {d} u_{3}.$ For example, in spherical coordinates $\mathrm {d} V=u_{1}^{2}\sin u_{2}\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}\,\mathrm {d} u_{3}$ , and so $\rho =u_{1}^{2}\sin u_{2}$ .

The concept and rule of the volume element is not limited to the spatial coordinate system or three dimensions: in two dimensions, it is also known as a topic called the area element, and in this setting it is useful for performing tasks such as surface integrals. Under the change of coordinates, the volume element is changed by the absolute value of the Jacobian determinant of the coordinate transformation (by the change of variables formula). This fact allows volume elements to be defined as a type of measure in a manifold. In an orientable differentiable manifold, a volume element usually arises from a volume form: the higher-order differential form. In a non-orientable manifold, the volume element is usually the absolute value of the (locally defined) volume form: it defines a density of a (locally defined) volume form: it defines a 1-density.

# Spatial geometry

Spatial geometry refers to Euclidean geometry in three-dimensional space. A space where height exists apart from length and width. Spatial geometry requires a lot of imagination. The whole world around us is three-dimensional and spatial. Any volume you know should have its properties calculated in the subject of spatial geometry. Shapes such as spheres, cones and cylinders are of this category. Spatial geometry includes three-dimensional spatial items (length, width, height).

Such as: area, volume, geometric volumes, polyhedra, conic section, three-dimensional space, spherical geometry, spherical coordinates, cylindrical coordinates and...

## History

The history of spatial geometry dates back to ancient Greece, the Pythagoreans dealt with regular solids, but the pyramid, prism, cone and cylinder were not studied until the Platonists. Eudoxus measured them and proved that the pyramid and the cone are one-third the volume of a prism and a cylinder on the same base and the same height. He probably also discovered the proof that the volume enclosed by a sphere is proportional to the cube of its radius.

## Definition of topics

### area and volume

Volume: The amount of space occupied by an object is called volume. The volume unit is equal to the cubic unit. Volume is a quantity of three-dimensional space that is limited by a specific boundary, for example, it is the space occupied by a substance (solid, gas, liquid, plasma) or its shape. Volume is a sub-unit of SI, which is the unit It is meter to the power of 3 (cubic meter). The volume of a container is equal to the volume of the liquid that fills it. To calculate the volume of certain 3D shapes, there are specific relationships that are simple relationships for simple shapes with geometric regularity. For complex shapes that do not have a simple relationship to calculate the volume, the volume can be obtained from integral methods. The volume of one-dimensional shapes, such as a line, or two-dimensional shapes, such as a plane, is zero.

Area: It is a type of quantity that calculates the surface area of three-dimensional objects and the internal value of two-dimensional objects. The area unit is equal to the square unit. Area is a quantity that expresses the extent of an area on a plane or on a curved surface. slow The area of the plane region or "plane area" refers to the area of a planar layer or layer, while "surface area" refers to the area of an open surface or boundary of a three-dimensional object. Area can be understood as the amount of material of a given thickness required to form a model of a shape, or the amount of paint required to cover a surface with a layer. This two-dimensional analog is the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).

### Geometric volumes and non-geometric volumes

Non-geometric volumes are complex volumes whose volumes are hard to obtain. But their area can be obtained, but it is a bit complicated. To obtain non-geometric volumes, we first pour water in a beaker. After we fill it with water and measure the amount of liters, we drop the non-geometric object into water with this The water rises, then we subtract the amount of water that has risen with a non-geometric volume by the amount of water that was determined before, and then we measure and write its volume.

"Geometric volumes" = geometric volumes are objects for which we can write surface and volume formulas. We can find the volume of those geometric objects by pattern-finding method by analyzing and measuring the volume of the corresponding components. And by summarizing and formulating it, we can get the formula of its volume. To find its area, we first calculate the area of its components by analyzing and drawing the shape in a continuous and discrete way, and write its formula by analysis.

Example = sphere, pyramid, prism, polyhedron, cylinder, cone and cube, tetrahedron, parallelogram

### definition of prism, sphere, pyramid, polyhedron

"Definition of Prism": A prism is a volume that has two bases, lateral faces, vertices and edges. The faces of a prism are rectangular and the number of its faces is equal to the number of sides of its base, the number of its vertices is twice as many as the faces and The number of edges is three times the face of the prism. The faces of the pyramid are obtained by the formula n+2, because the number of faces of the prism hub is always two more than the side face, because the other two faces are the base of the prism. In geometry, a prism is a polyhedron with a base of n- side, transferred base polygon (in another plane) and n other faces which are necessarily all parallelograms and connect the corresponding vertices of two n-gons. All cross sections parallel to the base are the same. Prisms are named according to the number of their base sides; So, for example, a prism with a pentagonal base is called a pentagonal prism. The definition of a prism to a pyramid is that a prism is the same as a pyramid, but its apex is at infinity.

Definition of a pyramid: A pyramid is a volume whose faces intersect at a point and whose faces are triangular with a base. The number of edges of the pyramid is twice the number of the sides of the base. In fact, a pyramid is a three-dimensional shape that is formed by connecting a point in space to all closed points on the plane. That point is called the top of the pyramid and that flat shape is called the base of the pyramid. The base of the pyramid is an arbitrary polygon and the other faces are equilateral triangles that connect to each other at the vertex. The vertical line that connects the vertex to the base is called the height of the pyramid. Among the most famous structures in the world in the form of a pyramid, we can mention the Egyptian triple pyramids.

"Definition of a sphere": A sphere is a perfectly round geometric object in three-dimensional space. For example, a ball is a sphere. A sphere, like a circle in two dimensions, is perfectly symmetrical around a point in three-dimensional space. All points on the surface of the sphere are at the same distance from the center of the sphere. The distance of these points from the center of the sphere is called the radius of the sphere and is represented by the letter "r". The longest distance from both sides of the sphere (which passes through the sphere) is called the diameter of the sphere. The diameter of the sphere also passes through its center and therefore its size is twice the radius. A sphere is a set of points in space that has a circular base and radius, which is a regular polyhedron. The sphere is the result of the period of a semicircle and a circle around the diameter, which rotates 180 degrees in a circle and 360 degrees in a semicircle. We divide the faces of the sphere into several degrees based on the division of its area, which is 360 degrees.

Polyhedron Definition: A polyhedron is a solid geometric object in three-dimensional space that has smooth faces (each face in one plane) and sides or edges located on a straight line. So far, no single definition has been provided for it. A tetrahedron is a type of pyramid and a cube is an example of a hexagon. A polyhedron can be convex or non-convex. Polyhedrons such as pyramids and prisms can be made by extruding two-dimensional polygons. There can only be a finite number of convex polyhedra with regular faces and equiangular shapes, including Platonic solids and Archimedean solids. Some Archimedean solids can be made by cutting the top pyramid of Platonic solids. Due to the simplicity of construction, polyhedra are used in most architectural works such as geodesic domes and pyramids. Recently, due to the use of shapes, interest in multifaceted surfaces has increased. Some compact molecules and atoms, especially crystalline structures and Platonic hydrocarbons, as well as some radials have a shape similar to Platonic solids. Platonic solids are also used in making dice. Polyhedra have different characteristics and types and are placed in different symmetry groups. Other polyhedra can be created by operations on any polyhedra. Some of them have relationships with each other. Polyhedra have been of interest since the Stone Age. The sphere is also considered as a family of polyhedra. Cube, tetrahedron, parallelogram are geometric volumes that are also considered polyhedra.

### conic section

In mathematics, a conic section (or simply a conic, sometimes called a quadratic curve) is a curve obtained as the intersection of the surface of a cone with a plane. Three types of conic sections are hyperbola, parabola and ellipse. The circle is a special case of the ellipse, although historically it is sometimes called the fourth type. Ancient Greek mathematicians studied conic sections, culminating in Apollonius Perga's systematic work on their properties around 200 BC.

### 3D space

In mathematics, "3D space" is a vector space with three dimensions and a geometric model of the physical world in which we live. The three dimensions are commonly known as length, width, and height (or depth), although this naming is optional.

### Spherical geometry

Spherical geometry is the branch of geometry that deals with the two-dimensional surface of a sphere. This is an example of geometry unrelated to Euclidean geometry. The practical application of spherical geometry is in the field of aviation and astronomy. In Euclidean geometry, straight lines and points are the main concepts. In Korea, dots are defined in their usual meaning. In Euclidean geometry, lines do not mean a straight line, but in the concept of the shortest distance between two points, a straight line is proposed, which is called a geodesic. On a sphere, geodesics are great circles. Other geometric concepts are defined on the page, except that a straight line is used instead of a great circle. Therefore, in spherical geometry, angles are defined between great circles, and as a result, spherical trigonometry is different from ordinary trigonometry in many ways. For example: the sum of the internal angles of a triangle is more than 180 degrees. Spherical geometry is not elliptic (Riemannian) geometry, but this feature that a line from a point cannot have a line parallel to it is common to both. In isometrics of spherical geometry with Euclidean geometry, the line from a point has a line parallel to itself, and in isometry with hyperbolic geometry, the line from a point has two lines parallel to itself and infinity. Concepts of spherical geometry may be applied to the spindle sphere, although slight modifications must be made to certain formulas.

### spherical coordinates

In mathematics, spherical coordinates are for three-dimensional space, in which the position of a point is determined by three numbers: the "radial distance" of that point from a fixed origin, "its polar angle measured from a direction," the apex fixed, and its orthogonal "orthogonal" angle on a reference plane that passes through the origin and is perpendicular to the vertex, is measured from a fixed reference direction in that plane. It can be seen as a three-dimensional version of the polar coordinate system.

The use of symbols and the order of coordinates are different in sources and disciplines. This paper uses the ISO convention often encountered in physics: it shows the radial distance, the polar angle, and the azimuth angle. In many math books, the radial distance shows the azimuthal angle and the polar angle and changes the meanings of "θ" and "φ". Other conventions are used, such as r for the radius from the z axis, so great care must be taken to check the meaning of the symbols.

According to the conventions of geographic coordinate systems, positions are measured by longitude and latitude and altitude (elevation). There are a number of celestial coordinate systems based on different base planes and with different terminology for different coordinates. Spherical coordinate systems used in mathematics usually use radians instead of degrees and measure the azimuthal angle counterclockwise from the x-axis to the y-axis rather than clockwise from north (0 degrees) to east. (90 degrees) like the horizontal coordinate system. . The polar angle is often replaced by the elevation angle measured from the reference plane, so that the zero elevation angle is at the horizon.

The spherical coordinate system is a generalization of the two-dimensional polar coordinate system. It can also be extended to higher dimensional spaces and then it is called a hyperspherical coordinate system.

### cylindrical coordinates

Cylindrical coordinate is a type of orthogonal coordinate in which a point is considered in space on the base of a cylinder. The location of that point is expressed based on the radius and height of the cylinder (r and z) and the angle that the radius of the base passing through that point makes with the x axis (θ). This device, in two-dimensional mode, is converted to polar coordinates by removing z. In physics and especially in the topics of electromagnetics and telecommunications, instead of r, θ, z, the letters ρ, φ, z are used respectively.

# Differential geometry

Differential geometry is a branch of mathematics that deals with the geometry of smooth shapes and smooth spaces, otherwise called smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multi-linear algebra. This field has its roots in the study of spherical geometry since ancient times. It is also related to astronomy, geodesy of the earth and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of flat spaces are curves and plane surfaces in Euclidean three-dimensional space, and the study of these forms was the basis for the development of modern differential geometry during the 18th and 19th centuries. A triangle immersed in a saddle-shaped plane (a hyperbolic parabola), as well as two divergent hyperparallel lines. Since the late 19th century, differential geometry has evolved into a field generally concerned with geometric constructions on differentiable manifolds. A geometric structure is a structure that defines a concept of size, distance, shape, volume, or other rigid structure. For example, in Riemannian geometry distances and angles are specified, in symplectic geometry volumes may be calculated, in isomorphic geometry only angles are specified, and in gauge theory certain fields are given on space. Differential geometry is closely related to, and sometimes includes, differential topology, which concerns properties of differentiable manifolds that do not rely on any additional geometric structure (see that article for further discussion of the distinction between the two ). Differential geometry is also related to the geometric aspects of the theory of differential equations, which is also called geometric analysis. Differential geometry is used throughout mathematics and the natural sciences. The language of differential geometry was further used by Albert Einstein in his theory of general relativity and subsequently by physicists in the development of quantum field theory and the Standard Model of particle physics. Outside of physics, differential geometry is used in chemistry, economics, engineering, control theory, computer graphics and computer vision, and more recently in machine learning.

# Area and volume

Area and volume is a topic of spatial geometry that deals with the properties, characteristics, application and calculation of volume and area of three-dimensional geometric volumes. Three-dimensional volumes are objects that have three dimensions (length, width, height).

Rotation, sections, cuts, three-dimensional drawing, extensive drawing of volumes, enclosure, volume and area are important elements of this topic.

Geometric volumes are divided into two categories:

1. Geometric volumes such as prisms, spheres, pyramids, etc.
2. Non-geometric volumes

## Definitions

### Definition of area and volume

Volume: The amount of space occupied by an object is called volume. The volume unit is equal to the cubic unit. Volume is a quantity of three-dimensional space that is limited by a specific boundary, for example, it is the space occupied by a substance (solid, gas, liquid, plasma) or its shape. Volume is a sub-unit of SI, which is the unit It is meter to the power of 3 (cubic meter). The volume of a container is equal to the volume of the liquid that fills it. To calculate the volume of certain 3D shapes, there are specific relationships that are simple relationships for simple shapes with geometric regularity. For complex shapes that do not have a simple relationship to calculate the volume, the volume can be obtained from integral methods. The volume of one-dimensional shapes, such as a line, or two-dimensional shapes, such as a plane, is zero.

Area: It is a type of quantity that calculates the surface value of three-dimensional objects and the internal value of two-dimensional objects. The area unit is equal to the square unit. Area is a quantity that expresses the extent of an area on a plane or on a curved surface. The area of the plane region or "plane area" refers to the area of a planar layer or layer, while "surface area" refers to the area of an open surface or boundary of a three-dimensional object. Area can be understood as the amount of material of a given thickness required to form a model of a shape, or the amount of paint required to cover a surface with a layer. This two-dimensional analog is the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).

### Definition of geometric and non-geometric volumes

Non-geometric volumes=Non-geometric volumes are complex volumes whose volumes are hard to obtain. But their area can be obtained, but it is a bit complicated. To obtain non-geometric volumes, we first pour water in a beaker. After we fill it with water and measure the amount of liters, we drop the non-geometric object into water with this The water rises, then we subtract the amount of water that has risen with a non-geometric volume by the amount of water that was determined before, and then we measure and write its volume.

Geometric volumes= geometric volumes are objects for which we can write surface and volume formulas. We can find the volume of those geometric objects by pattern-finding method by analyzing and measuring the volume of the corresponding components. And by summarizing and formulating it, we can get the formula of its volume. To find its area, we first calculate the area of its components by analyzing and drawing the shape in a continuous and discrete way, and write its formula by analysis.

Examples = sphere, pyramid, prism, polyhedron, cylinder, cone and cube, tetrahedron, parallelogram

### Notes on geometric volumes

Note 1: A regular polyhedron (hexahedral) cube has a square face that has two square bases, so the cube is a regular prism-polyhedron volume.

Note 2: A tetrahedron is a pyramid and a polyhedron with the base and sides of an equilateral triangle. So, a tetrahedron is considered a pyramid-polyhedral volume and a kind of Platonic solid.

Point 3: A parallelogram is a prismatic volume with a lateral face and is a hexagon with parallel faces. Therefore, a parallelogram is a prismatic-polyhedral volume.

### Definition of prism, sphere and pyramid

Definition of a prism: A prism is a volume that has two bases, lateral faces, vertices and edges. The faces of the prism are rectangular and the number of faces is equal to the number of sides of the base, the number of vertices is twice as many as the faces and the number of edges is three times the faces of the prism. The faces of the pyramid are obtained by the formula n+2, because the number of faces of the prism hub is always two more than the side faces, because the other two faces are the base of the prism. in another plane) and n other faces that are necessarily all parallelograms and connect the corresponding vertices of two n-gons. All cross sections parallel to the base are the same. Prisms are named according to the number of their base sides; So, for example, a prism with a pentagonal base is called a pentagonal prism. The definition of a prism to a pyramid is that a prism is the same as a pyramid, but its apex is at infinity.

Definition of a pyramid: A pyramid is a volume whose faces intersect at a point, and whose faces are triangular in shape with a base. The number of sides is the base. In fact, a pyramid is a three-dimensional shape that is created by connecting a point in space to all closed shape points on the plane. That point is called the top of the pyramid and that flat shape is called the base of the pyramid. The base of the pyramid is an arbitrary polygon and the other faces are equilateral triangles that connect to each other at the vertex. The vertical line that connects the vertex to the base is called the height of the pyramid. Among the most famous structures in the world in the form of a pyramid, we can mention the Egyptian triple pyramids.

Definition of a sphere: A sphere is a completely round geometric object in three-dimensional space. For example, a ball is a sphere. A sphere, like a circle in two dimensions, is perfectly symmetrical around a point in three-dimensional space. All points on the surface of the sphere are at the same distance from the center of the sphere. The distance of these points from the center of the sphere is called the radius of the sphere and is represented by the letter r. The longest distance from both sides of the sphere (which passes through the sphere) is called the diameter of the sphere. The diameter of the sphere also passes through its center and therefore its size is twice the radius. A sphere is a set of points in space that has a circular base and radius, which is a regular polyhedron. The sphere is the result of the period of a semicircle and a circle around the diameter, which rotates 180 degrees in a circle and 360 degrees in a semicircle. We divide the faces of the sphere into several degrees based on the division of its area, which is 360 degrees.

### Definition of cylinder, cone and polyhedron

Definition of a cylinder: A cylinder is a prismatic volume whose base is circular. In geometry, a cylinder is a spatial curved base whose surface is formed by a set of points. The edges of a cylinder are uncertain because its base is circular. is, it can be said that the side face, face, vertex, edge of the cylinder are 3n, 2n, n+2, n in order. The cylinder in differential geometry is drawn as a surface whose generator is a set of parallel lines. The definition of the cylinder in the cone of this The cylinder is the same as the cone, but its apex is at infinity. The cylinder is the result of the rotation of a rectangle around one of its sides (length, width) equal to 360 degrees.

Definition of a cone: A cone is a pyramidal volume whose base is circular, a cone is a three-dimensional geometric shape that slowly or quickly (depends on the base surface and height) from its flat base (cone cross-section) to the top. It narrows. More specifically, it is a solid shape bounded by a base plane (cone cross-section), and its lateral surface is the locus of straight lines connecting the tip of the cone to points around the base (cross-section). The word cone is sometimes referred to the top of this solid body and sometimes only to its lateral surface. Cones can be upright or slanted. It is necessary to mention that the volume of an oblique cone with a certain cross-sectional area and a certain height is equal to the volume of a right cone with the same area and a certain height. The cone resulting from the rotation of a right triangle around one of its adjacent sides is 360 degrees.

Definition of Polyhedron: A polyhedron is a solid geometric object in three-dimensional space that has flat faces (each face in one plane) and sides or edges located on a straight line. So far, no single definition has been provided for it. A tetrahedron is a type of pyramid and a cube is an example of a hexagon. A polyhedron can be convex or non-convex. Polyhedrons such as pyramids and prisms can be made by extruding two-dimensional polygons. There can only be a finite number of convex polyhedra with regular faces and equiangular shapes, including Platonic solids and Archimedean solids. Some Archimedean solids can be made by cutting the top pyramid of Platonic solids. Due to the simplicity of construction, polyhedra are used in most architectural works such as geodesic domes and pyramids. Recently, due to the use of shapes, interest in multifaceted surfaces has increased. Some compact molecules and atoms, especially crystalline structures and Platonic hydrocarbons, as well as some radials have a shape similar to Platonic solids. Platonic solids are also used in making dice. Polyhedra have different characteristics and types and are placed in different symmetry groups. Other polyhedra can be created by operations on any polyhedra. Some of them have relationships with each other. Polyhedra have been of interest since the Stone Age. The sphere is also considered as a family of polyhedra. Cube, tetrahedron, parallelogram are geometric volumes that are also considered polyhedra.

## area and volume of geometric shapes

Cube volume:$V=a^{3}\;$

Area of ​​the cube:$V=6a^{2}\;$

Tetrahedron volume:${\textstyle V={{\sqrt {2}} \over 12}a^{3}\,}$

Tetrahedron Area:$V={{\sqrt {3}}a^{2}\,}$

Volume of a regular octahedron:

$V={\frac {2}{3}}a^{3}$

Area of a regular octahedron:

$V={2a^{2}{\sqrt {3}}\,}$

Volume of a rectangular cube: ${\textstyle {V=abc}}$

Area of a rectangular cube: ${\textstyle A=2ab+2ac+2cb}$

The volume of the prism: $V=Sh$

The volume of a prism with a polygonal base:$V={\frac {n}{4}}ha^{2}\cot {\frac {\pi }{n}}$

The Area of a prism with a polygonal base:$A={\frac {n}{2}}a^{2}\cot {\frac {\pi }{n}}+nah$

cylinder volume: $V=\pi r^{2}h}$

cylinder Side area: $A=\ Ph+2S$

cylinder Area: $A=2\pi r(r+h)\,\!$

The volume of the pyramid: $V={\frac {1}{3}}Sh}$

The volume of the cone: $V={\frac {1}{3}}\pi r^{2}h$

The Area of the pyramid: $B+{\frac {PL}{2}}\,\!$

The Area of the cone: $\pi r(r+l)\,\!$

volume of sphere: $V={\frac {4}{3}}\pi r^{3}$

Area of sphere: $4\pi r^{2}\ {\text{or}}\ \pi d^{2}\,\!$

volume of Spherical: ${4 \over 3}\pi r^{2}h$

volume of Ovalcal: ${4 \over 3}\pi abc$

Area of Spherical=${A=4\pi ab}$

Area of Ovalcal=:$A=2\pi a^{2}\left(1+{\frac {c}{ae}}\arcsin e\right)$

The volume of the incomplete pyramid:$V={\tfrac {1}{3}}h\left(a^{2}+ab+b^{2}\right).$

The area of ​​the incomplete pyramid:$A={\frac {n}{4}}\left[\left(a_{1}^{2}+a_{2}^{2}\right)\cot {\frac {\pi }{n}}+{\sqrt {\left(a_{1}^{2}-a_{2}^{2}\right)^{2}\sec ^{2}{\frac {\pi }{n}}+4h^{2}\left(a_{1}+a_{2}\right)^{2}}}\right]$

The volume of the incomplete cone:$V={\tfrac {\pi }{3}}h\left(r^{2}+rr'+r'^{2}\right).}$

Area of ​​incomplete cone:{\begin{aligned}{\text{Total surface area}}&=\\&=\pi \left(\left(r_{1}+r_{2}\right){\sqrt {\left(r_{1}-r_{2}\right)^{2}+h^{2}}}+r_{1}^{2}+r_{2}^{2}\right)\end{aligned}}}

Torus volume:$V=2\pi ^{2}Rr^{2}=\left(\pi r^{2}\right)\left(2\pi R\right).\,$

Torus area:$A=4\pi ^{2}Rr=\left(2\pi r\right)\left(2\pi R\right)\,$

The Volume of ​​the parallelogram:$V=abc{\sqrt {K}}$

The area of ​​the parallelogram:$2{(ah+bh'+ch'')}$

Area of ​​a regular polyhedron:

$A=n({\tfrac {1}{4}}n'a^{2}\cot {\frac {\pi }{n'}})$

Volume of polyhedral solids:${\frac {1}{3}}\left|\sum _{F}(Q_{F}\cdot N_{F})\operatorname {Area} (F)\right|,$

## SA:V ratio of geometric volumes

The ratio of surface area to volume or the ratio of surface to volume, which is indicated by different signs such as sa/vol and SA:V; It is the amount of surface area per volume unit of an object or a set of objects. In chemical reactions where a solid substance is involved, the surface-to-volume ratio is an important factor that indicates that chemical reactions are taking place. The surface-to-volume ratio or SA:V is a formula that is the ratio of volume to total surface area, and its values are in Geometric volumes are different. The SA:V ratio depends on the size of the radius or the size of the geometric volume.

### proportion V/S geometric objects

Ratio of V/Scube:${\frac {a}{6}}$

Tetrahedral V/S ratio: ${\frac {{{\sqrt {2}} \over 12}a^{3}}{2a^{2}{\sqrt {3}}}}$

V/S ratio: ${\frac {Sh}{Ph+2s}}$

V/S ratio of cylinder: ${\frac {\pi r^{2}h}{2\pi r^{2}+2\pi rh}}$

V/S ratio of the pyramid: ${\frac {{\frac {1}{3}}Sh}{{\frac {N}{2}}Bh+S}}$

Cone V/S ratio:${\frac {{\frac {1}{3}}\pi r^{2}h}{\pi r^{2}+\pi rL}}$

The ratio V/S sphere=${\frac {r}{3}}$

### SA:V for regular balls and N next A plot of the surface-to-volume ratio (SA:V) value for a three-dimensional sphere showing increasing The radius of the ball has an inverse relationship with the ratio.

A ball is a three-dimensional object in the shape of a sphere (in this topic, most of the area (area) on the sphere is desired, not the volume inside it). Balls can exist in as many dimensions as needed and are generally called n-dimensional balls, where n is the number of dimensions of the ball. For a typical three-dimensional ball, SA:V can be calculated using the standard equation for area and volume; where the area $isS=4\pi r^{2}$  and the volume $isV=(4/3)\pi r^{3}$ . For a ball with unit radius (r=1), the ratio of surface to volume is equal to 3. SA:V has the opposite relationship with the radius, if the radius is doubled, the SA:V is halved.

The above argument can be extended for the n-dimensional ball and the general relationships of volume and surface area can be written as follows:

$V={\frac {r^{n}\pi ^{\frac {n}{2}}}{\Gamma (1+n/2)}}$  Volume;$S={\frac {nr^{n-1}\pi ^{\frac {n}{2}}}{\Gamma (1+n/2)}}$  surface area

The ratio ${\frac {V}{S}}$  is reduced to $nr^{-1}$  in the next n state; So the same linear relationship holds for area and volume in every dimension: doubling the radius always halves the ratio.

## Proof of formulas of geometric volumes

These topics are a partial proof of the list and a reference to the proof of their formulas. To see the full proof, you should read the sections on three-dimensional geometric shapes in the section on geometry in supplementary mathematics.

### cube

The volume of a cube acts like the volume of a rectangular cube, but because the sides of the cube are equal, it is placed as a power of three on its side.

$V=S.H=(a^{2}).a=a^{3}$

The area of ​​the cube is also proven in different ways, as we know that the cube has six square faces that are regular quadrilaterals, and its total area is obtained based on the sum of the six square faces of the cube.The total area of ​​the cube, like the volume of the cube, is used through prism formulation and is proved according to the total area of ​​the prism

$V=P.H+2S=4a^{2}+2a^{2}=6a^{2}$

### Parallelepiped

The volume of the parallelogram is obtained based on three vectors a, b, c, which are determined in the form of determinants. But first, it should be calculated trigonometrically and then written as a determinant.

$S=\left|\mathbf {a} \right|\cdot \left|\mathbf {b} \right|\cdot \sin \gamma =\left|\mathbf {a} \times \mathbf {b} \right|$

$h=\left|\mathbf {c} \right|\cdot \left|\cos \theta \right|$

After the calculation, we do the work in the form of determinants.

$V=\left|\det {\begin{bmatrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{bmatrix}}\right|.$

• ${\vec {a}}=(a_{1},a_{2},a_{3})$
• ${\vec {b}}=(b_{1},b_{2},b_{3})$
• ${\vec {c}}=(c_{1},c_{2},c_{3})$

The volume of the parallelogram is like this.

$V=|({\vec {a}}.{\vec {b}}).{\vec {c}}|=a_{1}b_{1}c_{1}+a_{2}b_{2}c_{2}+a_{3}b_{3}c_{3}$

### prism

The volume of the prism, if s is the area of ​​the base and h is the height, its volume is: $V=Sh$

If p is the perimeter of the base and h is the height, the lateral area of ​​the prism is written accordingly.

$S=Ph$

The total area of ​​the prism can be written based on this formula if s is the area of ​​the base $S=Ph+2s$

### cylinder

A cylinder is one of the basic curved shapes in geometry, whose outer surface is a set of points that are at the same distance from a straight line. The name of this line is the right axis. The two ends of this spatial shape are blocked by two plates perpendicular to the axis of the closed cylinder. The surface and volume of the cylinder have been known to mathematicians since the distant past.

The volume of the cylinder is calculated based on the volume of the prism

$V=S.H=\pi R^{2}H$

# Regular polygon

"Regular polygon" in Euclidean geometry is a polygon whose angles and sides are equal. Regular polygons can be cuboid or star-shaped. In the limiting case, a sequence of regular polygons with increasing number of sides becomes a circle if the perimeter remains constant and becomes an apeirogon if the length of the side remains constant.

## Perimeter and Area

### perimeter

The perimeter of a regular polygon is based on the multiplication of the number of sides of a regular polygon and the size of the sides of a regular polygon.

The perimeter of the area is obtained based on this relationship. $P=a_{1}+a_{2}+...+a_{n}=(n.a)=na$  Here n is equal to the number of sides of a regular polygon and a here is equal to the size of the sides of a regular polygon Is.

### Area

The area of a regular polygon is obtained based on trigonometric relationships. The area of a regular polygon is based on the fact that it is made of 1x1 squares and the number of its sides is n, and because it is based on the number pi, the number of sides expands trigonometrically based on the cotangent in the form of pi. It is obtained by dividing the number of sides of a regular polygon.

The area of a regular polygon is written accordingly:

$A={\frac {1}{4}}n.a^{2}\cot({\frac {\pi }{n}})$

Here, pi is in radians (equal to 180°).

#### The square area relationship by trigonometric method

Since the square is a regular polygon, its area can also be written as the area of a regular polygon which is obtained by trigonometric method, which is as follows: ${\frac {1}{4}}na^{2}\cot {\frac {\pi }{n}}=a^{2}\cot {\frac {\pi }{4}}=a^{2}.({\sqrt {1}})=1.a^{2}=a^{2}$  here:

${\frac {1}{4}}na^{2}={\frac {1}{4}}4a^{2}=a^{2}$

$\cot {\frac {\pi }{4}}=1$

Therefore, the area of a parallel square is equal to the square of its side.

#### Other other formula area

The area of an "n-" regular polygon with the size of the side a, the radius of the surrounding circle "R", the radius of the surrounding circle "r" and the perimeter "p" is obtained using the following relations:

$A={\tfrac {1}{2}}nar={\tfrac {1}{2}}pr={\tfrac {1}{4}}na^{2}\cot {\tfrac {\pi }{n}}=nr^{2}\tan {\tfrac {\pi }{n}}={\tfrac {1}{2}}nR^{2}\sin {\tfrac {2\pi }{n}}$

where R is equal to:

$R={\frac {s}{2\sin \left({\frac {\pi }{n}}\right)}}={\frac {a}{\cos \left({\frac {\pi }{n}}\right)}}$

# Conical section

In mathematics, a conic section (or simply a conic, sometimes called a quadratic curve) is a curve obtained as the intersection of the surface of a cone with a plane. Three types of conic sections are: hyperbolic, parabolic and elliptical. A circle is a special case of an ellipse, although historically it is sometimes called the fourth type. Ancient Greek mathematicians studied conic sections, culminating in Apollonius Perga's systematic work on their properties around 200 BC.

## Definition

### Circle

A circle is the set of points in a plane that are equidistant from a given point O . The distance  from the center is called the radius, and the point O is called the center. Twice the radius is known as the diameter D=2r . The angle a circle subtends from its center is a full angle, equal to 360° or $2\pi$  radians.

A circle has the maximum possible area for a given perimeter, and the minimum possible perimeter for a given area.

### Ellipse

In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity $e$ , a number ranging from $e=0$  (the limiting case of a circle) to $e=1$  (the limiting case of infinite elongation, no longer an ellipse but a parabola).

### Parabola

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

The parabola was studied by Menaechmus in an attempt to achieve cube duplication. Menaechmus solved the problem by finding the intersection of the two parabolas $x^{2}=y$ and$y^{2}=2x$  . Euclid wrote about the parabola, and it was given its present name by Apollonius. Pascal considered the parabola as a projection of a circle, and Galileo showed that projectiles falling under uniform gravity follow parabolic paths. Gregory and Newton considered the catacaustic properties of a parabola that bring parallel rays of light to a focus (MacTutor Archive), as illustrated above.

### Hyperbola

Each branch of the hyperbola has two arms which become straighter (lower curvature) further out from the center of the hyperbola. Diagonally opposite arms, one from each branch, tend in the limit to a common line, called the asymptote of those two arms. So there are two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the mirror point about which each branch reflects to form the other branch. In the case of the curve  the asymptotes are the two coordinate axes.

Hyperbolas share many of the ellipses' analytical properties such as eccentricity, focus, and directrix. Typically the correspondence can be made with nothing more than a change of sign in some term. Many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids (saddle surfaces), hyperboloids ("wastebaskets"), hyperbolic geometry (Lobachevsky's celebrated non-Euclidean geometry), hyperbolic functions (sinh, cosh, tanh, etc.), and gyrovector spaces (a geometry proposed for use in both relativity and quantum mechanics which is not Euclidean).

## Equations of Conics

The conic sections can be described in a suitable x-y coordinate system by 2nd degree equations:

• Ellipse with center M at point (0,0) and major axis on x-axis:
${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1,\quad b=|MS_{3}|,\qquad a,b\neq 0\quad ,$  (see image). (For $a=b=r$  there is a circle.)
• Parabola with vertex at point (0,0) and axis on y-axis:
$y=ax^{2},\quad a={\frac {1}{4|SF|}},\qquad a\neq 0\quad ,$  (see figure).
• Hyperbola with center M at point (0,0) and major axis on x-axis:
${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1,\quad b^{2}=|MF_{1}|^{2}-a^{2},\ qquada,b\neq 0\quad ,$  (see figure).
• Intersecting pair of lines with intersection at point (0,0):
$a^{2}x^{2}-y^{2}=0,\ a\neq 0.$
• Line through the point (0,0):
$x^{2}=0.$
• point, the point (0,0):
$a^{2}x^{2}+b^{2}y^{2}=0,\ a,b\neq 0.$

For the sake of completeness, two more cases are added, which do not appear as actual conic sections, but are also described by equations of the 2nd degree:

• Parallel pair of lines:
$x^{2}=a^{2},\ a\neq 0.$
• The empty set:
$x^{2}+y^{2}=-1$  or $x^{2}=-1$ .

The last two cases can appear as plane sections of a right circular cylinder. A circular cylinder can be viewed as the limiting case of a cone with a cone apex at infinity. Therefore these two cases are included in the conic sections.

## Plane sections of the unit cone

conic-cases To determine that the curves/points referred to above as conic sections actually occur when a cone intersects a plane, here we intersect the unit cone (straight circular cone) $K_{1}\colon x^{2}+y^{2}=z^{2}$  with a plane parallel to the y-axis. This is not a limitation as the cone is rotationally symmetrical. Any right circular cone is the affine image of the unit cone $K_{1}$  and ellipses/hyperbolas/parabolas/... go back into the same with an affine mapping.

Given: plane $\varepsilon \colon ax+cz=d\ ,$  cone $K_{1}\colon x^{2}+y^{2}=z^{2}$ .

Wanted: Intersection $\varepsilon \cap K_{1}$ .

• Case I: $c=0$  In this case the plane is perpendicular and $a\neq 0$  and $x=d/a$  . Eliminating $x$  from the cone equation yields $z^{2}-y^{2}=d^{2}/a^{2}$ .
• Case Ia: $d=0$ . In this case the intersection consists of the pair of lines $t(0,1,\pm 1),\ t\in \mathbb {R} .$ .
• Case Ib: $d\neq 0$ . The above equation now describes a hyperbola in the y-z plane. So the intersection curve $\varepsilon \cap K_{1}$  is itself a hyperbola.
• Case II: $c\neq 0$ . If you eliminate $z$  from the cone equation using the plane equation, you get the system of equations $(1)\quad (c^{2}-a^{2})x^{2}+2adx+c^{2}y^{2}=d^{2},\qquad (2)\quad ax+cz=d.$
• Case IIa: For $d=0$  the plane goes through the apex of the cone $(0,0,0)$  and equation (1) has now the form $(c^{2}-a^{2})x^{2}+c^{2}y^{2}=0$ .
For $c^{2}>a^{2}$  the intersection is the point $P_{0}=(0,0,0)$ .
For $c^{2}=a^{2}$  the intersection is the line $t(c,0,-a),\ t\in \mathbb {R} .$
For $c^{2}  the intersection is the pair of lines $t(c/\pm {\sqrt {a^{2}-c^{2}}},1,-a/\pm {\sqrt {a^{2}-c^{2}}}),\ t\in \mathbb {R} .$
• Case IIb: For $d\neq 0$  the plane does not go through the apex of the cone and is not perpendicular.
For $c^{2}=a^{2}$ , (1) goes into $x=-{\frac {c^{2}}{2ad}}y^{2}+{\frac {d}{2a}}$  across and the intersection curve is a parabola.
For $c^{2}\neq a^{2}$  we transform (1) into ${\frac {(c^{2}-a^{2})^{2}}{d^{2}c^{2}}}\left(x+{\frac {ad}{c^{2}-a^{2}}}\right)^{2}+{\frac {c^{2}-a^{2}}{d^{2}}}y^{2}=1$  .
For $c^{2}>a^{2}$  the intersection curve is an ellipse and
for $c^{2}  there is a hyperbola.

Parametric representations of the intersection curves can be found in Weblink CDKG, pp. 106-107.

Summary:

• If the cutting plane does not contain the apex of the cone, the non-degenerate conic sections result (see figure for Ib, IIb), namely a parabola, an ellipse or a hyperbola , depending on whether the axis of the cone is intersected by the cutting plane at the same, greater, or lesser angle than the generatrices of the cone.
• If, on the other hand, the apex of the cone is in the section plane, the degenerate conic sections are created (see picture for Ia, IIa), namely a point (namely the apex of the cone), a straight line ' (namely one surface line) or a intersecting pair of straight lines, (namely two surface lines).

## Pencil of conics

A (non-degenerate) conic is completely determined by five points in general position (no three collinear) in a plane and the system of conics which pass through a fixed set of four points (again in a plane and no three collinear) is called a pencil of conics. The four common points are called the base points of the pencil. Through any point other than a base point, there passes a single conic of the pencil. This concept generalizes a pencil of circles.

In a projective plane defined over an algebraically closed field any two conics meet in four points (counted with multiplicity) and so, determine the pencil of conics based on these four points. Furthermore, the four base points determine three line pairs (degenerate conics through the base points, each line of the pair containing exactly two base points) and so each pencil of conics will contain at most three degenerate conics.

A pencil of conics can be represented algebraically in the following way. Let C1 and C2 be two distinct conics in a projective plane defined over an algebraically closed field K. For every pair λ, μ of elements of K, not both zero, the expression:

$\lambda C_{1}+\mu C_{2}$

represents a conic in the pencil determined by C1 and C2. This symbolic representation can be made concrete with a slight abuse of notation (using the same notation to denote the object as well as the equation defining the object.) Thinking of C1, say, as a ternary quadratic form, then C1 = 0 is the equation of the "conic C1". Another concrete realization would be obtained by thinking of C1 as the 3×3 symmetric matrix which represents it. If C1 and C2 have such concrete realizations then every member of the above pencil will as well. Since the setting uses homogeneous coordinates in a projective plane, two concrete representations (either equations or matrices) give the same conic if they differ by a non-zero multiplicative constant.

# Riemannian geometry

Riemannian geometry is a branch of differential geometry that investigates and studies the contents of Riemannian manifolds, i.e. smooth manifolds equipped with Riemannian metric, this manifold structure is equipped with inner multiplication on the tangent space at any point, so that from one point to The other point changes smoothly. Also, this structure specifically acquires local concepts such as angle, bend length, surface area and volume. From these, some other global quantities can be obtained by integration.

# Analytic geometry

Analytical geometry (also vector geometry) is a branch of geometry that provides algebraic tools (mainly from linear algebra) to solve geometric problems. In many cases, it makes it possible to solve geometric problems purely by calculation, without using the visual aid.

On the other hand, geometry that justifies its propositions on an axiomatic basis without reference to a number system is called synthetic geometry.

The methods of analytical geometry are used in all natural sciences, but above all in physics, such as in the description of planetary orbits. Originally, analytical geometry dealt only with questions of planar and spatial (Euclidean) geometry. In the general sense, however, analytic geometry describes affine spaces of any dimension over any body.

# Non-Euclidean geometry

Geometry is an area of mathematics that considers the regularities of position, size and shape of sets of points (e.g. on and between lines and surfaces), including their change and mapping. Depending on whether metric relationships (length, angular sizes, areas, volumes) are examined or whether only the mutual position of the objects is considered, one speaks of metric or projective geometry.

Metric geometries are Euclidean geometry, which is based on the parallel axiom, and the non-Euclidean geometries, such as Bolyai-Lobachevskian (hyperbolic) geometry, which retains all the axioms of Euclidean geometry but does not use the parallel axiom, and Riemannian (elliptical) Geometry that is also based on the assumption that not every straight line is infinitely long.

## geometries in the overview

Geometry is an area of mathematics that considers the regularities of position, size and shape of sets of points (e.g. on and between lines and surfaces), including their change and mapping. Depending on whether metric relationships (length, angular sizes, areas, volumes) are examined or whether only the mutual position of the objects is considered, one speaks of metric or projective geometry. Metric geometries are Euclidean geometry, which is built on the parallel axiom, and the non-Euclidean geometries, such as Bolay-Lobachevskian (hyperbolic) geometry, which retains all the axioms of Euclidean geometry but does not use the parallel axiom, and Riemannian (elliptical) Geometry that is also based on the assumption that not every straight line is infinitely long. The projective geometry can develop these three geometries as special forms of a general dimension geometry.

## on the emergence of non-Euclidean geometries

For around 2000 years, the general validity of Euclidean geometry was believed to exist, e.g. to describe the real physical space. But then the increasing criticism of this view led to two important discoveries:

The criticism of EUCLID's separation of geometry and arithmetic led to the creation of the concept of the real number, with the help of which not only commensurate but also incommensurable quantities could be characterized. The starting point for the emergence of a mathematics of constantly changing quantities was laid. RENÉ DESCARTES and CARL FRIEDRICH GAUSS worked in this field. Criticism of individual postulates, especially the fifth (parallel postulate), led to the development of other geometries that did not contradict reality - the non-Euclidean geometries (by LOBATSCHEWSKI, BOLYAI, GAUSS or RIEMANN), which led to the transition from a mathematics of constant relations to one of the mutable relationships meant. So the parallel axiom has been replaced by its opposite statement: "In a plane, through any point outside a given line, one can draw more than one line that does not intersect the given line". This geometry turned out to be just as consistent as Euclid-Hilbert's geometry. If many geometries are possible, there can be no general definition of the basic terms. The effort (of EUKLID and others) to define basic terms is therefore impossible in principle. Basic terms therefore only refer to the system under consideration.

## But which geometry is valid in reality?

The statement that Euclidean geometry is a model of the real space of our intuition does not answer this question. In experiments on small areas of the earth's surface, the assumption that the surface is a plane may apply. If, on the other hand, you are experimenting in large areas, you have to imagine the surface as curved or as a sphere. Correspondingly, the non-Euclidean geometries in small areas hardly differ from the Euclidean geometry. The difference only becomes apparent in large room areas. This question about the geometric structures of the real world led to new discoveries and developments in the natural sciences, such as B. the theory of relativity by EINSTEIN, which broke radically with usual geometric ideas.

## expansion

Neo-Euclidean geometry was later developed by Gauss and Riemann in the form of a more general geometry. This is the more general geometry that was used in Einstein's theory of general relativity. In non-Euclidean geometry, the set of interior angles is not like 180 degrees. For example, if the sides of the triangle are hyperbolic, the set of internal angles never reaches 180 degrees and is less. Also, if the geometry is elliptical, it will never be 180 degrees; Rather, it is more.

# Internal and external angle

An interior angle or interior angle of a polygon is another type of angle used to measure the interior angle of a regular polygon.

An exterior angle is another type of angle used to measure the exterior angles of regular polygons.

## Features

• In an interior angle, the greater the number of sides, the greater the size of the interior angle.
• In the exterior angle, the more the number of sides, the smaller the exterior angle.
• If we add the internal angle to the external angle, it becomes equal to 180 degrees.
• Internal angle and external angle are complementary to each other.
• The sum of interior angles depends on the number of sides of a regular polygon
• The sum of the external angles of a regular polygon is always equal to 360 degrees

An equilateral triangle is the only regular polygon whose exterior angle is greater than its interior angle.

• The sum of the interior angles of a triangle is 180 degrees.

A square is the only regular polygon whose interior angle is equal to its exterior angle.

## internal angle measurement

Consider a regular n-gon.

First, we calculate the number of triangles according to the famous polygons

• Square: 2 triangles
• Regular pentagon: 3 triangles
• Regular hexagon: 4 triangles
• Regular heptagon: 5 triangles
• Regular octagon: 6 triangles
• Regular octagon: 7 triangles
• Regular decagon: 8 triangles

According to this model, we find that the number of triangles is less than the number of sides of regular polygons. Therefore, the number of triangles inside any regular polygon is equal to this relationship.

Number of triangles: $n-2$

Because the set of interior angles of a triangle is 180 degrees, the set of interior angles is based on the sum of the angles of the number of triangles.

Sum of interior angles:$180(n-2)$

The interior angle measure of a regular polygon is equal to the number of divided sides. Because the number of vertices is equal to the number of sides.

Sum of Size of interior angles: ${\frac {180(n-2)}{n}}$

## external angle measurement

The sum of the external angles of any regular polygon is equal to 360 degrees. Therefore, to measure the exterior angle, we must divide 360 degrees by the number of sides of a regular polygon to determine the size of the angle.

Sum of external angles: 360 degrees: ${\frac {360}{n}}$

## Sum of internal angle measures of regular polyhedral faces

The sum of the internal angles of a polygon is calculated by the formula of the internal angles of the polygon itself, since its faces are regular polygons. We call this angle the interior angle of a polyhedron, which is a form of this relation.

Interior angle measure of a regular polyhedron:$n[(n'-2){\frac {180}{n'}}]$

blockquote where n is equal to the number of faces and n is the number of sides of a regular polyhedron.

## TABLE OF INTERIOR ANGLES

The name of the polygon Sum of interior angles The size of the interior angle External angle size
Equilateral triangle $180$  $60$  $120$
Square $360$  $90$  $90$
Regular pentagon $540$  $108$  $72$
Regular hexagon $720$  $120$  $60$
Regular octagon $1080$  $135$  $45$
Regular Nonagon $1260$  $140$  $40$
Regular decagon $1440$  $144$  $36$
Regular dodecahedron $1800$  $150$  $30$
Regular pentagon $2340$  $156$  $24$
Regular hexagon $2520$  $157.5$  $22.5$
regular dodecahedron $3240$  $162$  $18$
Twenty-four regular sides $3960$  $165$  $15$
Regular triangle $5040$  $168$  $12$
Thirty regular dodecahedrons $5400$  $168.75$  $11.25$
Thirty regular hexagons $6120$  $170$  $10$
Regular quadrilateral $6840$  $171$  $9$
Regular hexagon $10440$  $174$  $6$
Regular octagon $15840$  $176$  $4$
Regular centagon $17640$  $176.4$  $3.6$
One hundred and twenty regular polygons $21240$  $177$  $3$

# Cylindrical coordinate system

The distance from the selected reference plane that is perpendicular to the axis. Cylindrical coordinate system or abbreviated cylindrical coordinate system is a three-dimensional coordinate system and has calculations that use length, width, height, angles and rarely integral and trigonometric calculations. This type of coordinate system is simpler than spherical coordinates. To determine the position in cylindrical coordinates by measuring the point from the distance of a reference axis and the center as its distance and determining the dimension and direction of the axis point relative to where the point is located and the distance from the plane is determined. The selected reference perpendicular to the axis must be specified. It depends on which direction and at which point and in which sign the reference plane is located, and the distance is also determined as a positive and negative sign.The origin of the coordinates is also a place in all three longitudinal dimensions of the coordinates must be zero.The distance from the point to the axis is in the form of radius and length or in short height.Spherical and cylindrical coordinates have two sub-sets called longitudinal coordinates and angular coordinates, whose longitudinal coordinates are made in the form of threes based on integrals and trigonometry, and their angles are calculated in the form of spatial angle calculation. Of course, in coordinates Spherical, more integral and complex trigonometric calculation and spatial angle calculation form are performed than cylindrical coordinates.Cylindrical coordinate form is more algebraic and has a little bit of integral and trigonometry.The method of writing cylindrical coordinates is similar to three-dimensional coordinates, which starts with length, width, and height, respectively, from top to bottom.

# Cube

A cube is a three-dimensional closed volume that consists of 6 equal squares. In such a way that each side of each square is shared with only one other square and three squares are connected to each other at the vertices. The cube can be called a regular hexagon and is one of the five Platonic solids. If we change all or some of the faces of a cube from a square to a rectangle, the resulting hexagon is called a rectangular cube, and if we change its faces to rhombuses and parallelograms, it becomes a parallelogram. Sometimes, to distinguish it from a rectangular cube, a cube (with square faces) may also be called a square cube.In total, the cube has 2 bases, 4 sides, 6 faces, 8 vertices and 12 edges.

# Prism

In spatial geometry, a prism (which is derived from the Greek word πρίσμα, which means prisma) is a completely three-dimensional shape, and it is said to be a member of the polyhedron, which is made of two bases and faces. Prisms are known based on their bases. As a prism with a fourteen-sided base, it is called a fourteen-sided prism.

The prism was described for the first time by Euclid and in the 11th book of his geometry he described it like this (solid shapes formed by two opposite, equal and parallel planes are called prisms)

Of course, the definition of this type of theorem is extremely complicated to describe the prism, which has become the subject of confusion and controversy among geometers and subsequent geometers due to the lack of a rule definition.

# Cone

link=https://de.wikipedia.org/wiki/Datei:Cone_3d.png|thumb|Cones A cone is a geometric body that is created when all the points of a bounded and connected area lying in one plane are connected in a straight line with a point outside the plane. If the patch is a circular disk, the solid is called a circular cone. The area is called the base area, its boundary line is the directrix, the point is called the tip, apex or vertex of the cone and the area on the side is called the lateral surface. A cone has a vertex (the apex), an edge (the directrix) and two faces (the lateral and the base).

The vertex of a cone is not a vertex because the vertex is not an endpoint of edges (see definition of vertex).

The height of the cone means both the perpendicular from the tip to the base (i.e. the height is always perpendicular to the base) and the length of this perpendicular (i.e. the distance between the tip and the base).

The lines connecting the tip with the directrix are called generatrices, their union forms the cone envelope or the lateral surface. Ellipse, parabola and hyperbola are conic sections. In connection with conic sections, a "cone" is often understood as a "double cone".

Especially in technology, a cone or a truncated cone is often referred to as a cone (from Latin conus) or conical.

# Polyhedron

A polyhedron is a solid geometric object in three-dimensional space that has smooth and regular faces (each face in one plane) and sides or edges located on a straight line. So far, no single definition has been provided for it. A tetrahedron is a type of pyramid and a cube is an example of a hexagon. A polyhedron can be convex or non-convex. Polyhedrons such as pyramids and prisms can be made by extruding two-dimensional polygons. There can only be a finite number of convex polyhedra with regular faces and equiangular shapes, including Platonic solids and Archimedean solids. Some Archimedean solids can be made by cutting the top pyramid of Platonic solids. Due to the simplicity of construction, polyhedra are used in most architectural works such as geodesic domes and pyramids. Recently, due to the use of shapes, interest in multifaceted surfaces has increased. Some compact molecules and atoms, especially crystalline structures and Platonic hydrocarbons, as well as some radials have a shape similar to Platonic solids. Platonic solids are also used in making dice. Polyhedra have different characteristics and types and are placed in different symmetry groups. Other polyhedra can be created by operations on any polyhedra. Some of them have relationships with each other. Polyhedra have been of interest since the Stone Age. The sphere is also considered as a family of polyhedra. Cube, tetrahedron, parallelogram are geometric volumes that are also considered polyhedra.

## Definitions

Convex polyhedra are defined and convex polyhedra themselves are well-defined and can be calculated volume and area and can be used except for geometric volumes. But concave polyhedra are non-geometric volumes, and their definition is difficult and very difficult, and they do not have constant area and volume formulas. Geometric and non-geometric volumes are of the type of polyhedra, but their difference is in their concavity and convexity.

From these definitions, the following can be mentioned:

• A common and somewhat simple definition of a polyhedron is: a solid object whose outer surfaces can be covered with a large number of faces, or a solid formed by the union of convex polyhedra. A natural extension of this definition requires that the solid in question is bounded, its interior and possibly its boundary also connected. The faces of such a polyhedron can be defined as the connected space of the boundary parts inside each of the planes that cover it, and their sides and vertices as line segments and points where the faces meet. However, a polyhedron defined in this way does not include star-crossed polyhedrons whose faces may not form simple polygons and some of whose sides belong to more than two faces.
• Definitions based on the idea of a limiting surface rather than a solid are also common. For example, O'Rourke (1993) defines a polyhedron as a collection of convex polygons (its faces). These polygons are arranged in space in such a way that the intersection (or sharing) of both polygons is a common vertex or side or the null set, so that their union is a manifold. If a planar part of such a surface is not itself a convex polygon, O'Rourke stipulates that the part must be divided into pieces, each of which is a smaller convex polygon, such that the dihedral angles between them are flat. Somewhat more generally, Branko Grünbaum defines a polyhedron as a set of simple polygons that form an embedded manifold, with each vertex reached by at least three sides and any two faces only at vertices and sides common to each. They intersect. Cromwell's book on polyhedra gives a similar definition but without the restriction of at least three sides per vertex. Again, this type of definition does not include intersecting polyhedra. Similar concepts form the basis of topological definitions of polyhedra, as subsets of a topological manifold into topological disks (faces) whose binary intersections are points (vertices), topological arcs (sides). or the set is empty. However, there are topological polyhedra (even with perfect triangles) that cannot be understood as geometric polyhedra.
• A more modern definition based on the theory of single polyhedra is also prevalent. These polyhedra can be defined as sets with partial order, such that their elements are vertices, sides and faces of a polyhedron. When a vertex or side is smaller than a side or face, an element of the vertex or side is less than the element of the side or face (thus minor). Furthermore, there may be one special lower element of this partial order (denoting the null set) and one upper element representing the whole polyhedron. If the partial order segments between the elements of three faces apart (i.e., between each face and the bottom element and between the top element and each vertex) have the same structure as the abstract representation of a polygon, then these ordered sets contain exactly the same information as They carry topological polyhedra. However, these requirements are often permissive, instead requiring only that the cross-sections between elements of two faces from each other have the same structure as the abstract representation of a line segment. (This means that each side contains two vertices and belongs to two faces, and each vertex in a face belongs to two sides of that face.) A geometric polyhedron, defined in other ways, can be abstracted in this way. be described, but abstract polyhedra can also be used as a basis for defining geometric polyhedra. The realization of an abstract polyhedron is generally considered as a mapping of the vertices of the abstract polyhedron to geometric points, such that the vertices of each face are coplanar. So, a geometric polyhedron can be defined as the realization of an abstract polyhedron. Realizations that remove the flatness requirement, impose additional symmetry requirements, or map the vertices to higher dimensional spaces are also considered. The latter definition, in contrast to the definitions based on solids and processes, is quite suitable for star polyhedra. However, without additional restrictions, this definition allows the construction of polyhedra or unfaithful polyhedra (for example, by mapping all vertices to a single point) and the question: "How do we constrain the realization of some of these to avoid these polyhedra?" be prevented?" also remains unresolved.
• In all these definitions, a polytope can be understood as a three-dimensional instance of a more general polytope in any number of dimensions. For example, a polygon has a two-dimensional body and no faces, while a tetrapolytope has a four-dimensional body and an additional set of three-dimensional "cells". However, some texts on higher-dimensional geometry use the term "polytope" to mean something else: not a three-dimensional polytope, but a shape that is somehow different from a polytope. For example, some sources define a convex polygon as the intersection of many half-spaces and a polytope as a bounded polytope. In this article, only 3D polyhedra are discussed.

### Angles

Flat angle: Each of the corner angles of the faces of polyhedral polygons is called a flat angle. Space angle: Each of the angles that a polyhedron covers on a vertex in three-dimensional space is called a space angle. Each of these angles is bounded by three or more than three right angle angles. Dihedral angle: Any angle between two polyhedral faces is called a dihedral angle.

### polyhedral surface

A "polyhedral surface" is the result of joining a limited number of flat polygonal faces and does not necessarily enclose space. Polyhedral surfaces can have boundary sides and boundary vertices (only when the polyhedral surface contains only one face).

Recently, due to the use of shapes, the interest in multifaceted surfaces in architecture has increased.

## Preliminaries

Preliminaries are small things that we define simply.

### face

In Polyhedra, 'Face is any of Polygons having The area is"1" which forms part of the boundary of a solid body. A three-dimensional solid bounded exclusively by faces is a polyhedron.

In more technical methods of the geometry of polyhedra and higher-dimensional polytopes, the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions).

### vertex

"Vertex" (Arabic: "Ras") (English: vertex) in geometry is a point where two straight sides of an open or closed polygon meet. In other words, the vertex is the tip of the corners or the intersection of the lines of a geometric shape. A line is formed by connecting two vertices to each other, and a surface is formed by connecting three vertices to each other.

In 3D computer graphics models, vertices are usually used to define surfaces (usually triangles), and each vertex in these models is represented as a vector. In graph theory, vertices are also called nodes.

The number of vertices of any polygon in the plane is equal to the number of its sides.

### side

In geometry, a "side" or "line" or "edge" is a line segment that connects two adjacent vertices in a polygon; So in practice, a side is the interface for a one-dimensional line segment and two zero-dimensional objects.

Sides are the lines that make up each shape, and their number is often different in each shape compared to other shapes. For example, triangles have 3 sides and squares and rectangles have 4 sides.

A planar closed sequence of sides forms a polygon (and a face). In a polyhedron, exactly two faces touch each other on each side, while in higher dimensional polyhedra, three or more faces touch each other on each side.2 sup>

### angles

A flat angle is the corner angle of a polygonal face.

A solid angle is an angle in three-dimensional space that covers a polyhedron on a vertex. This angle is enclosed by three or more than 3 solid angles.

Dihedral angle is the angle between two adjacent faces

1. The lateral faces have a surface area

2. Contact means because they have a common vertex.

## FEATURES AND CHARACTERISTICS

### Number of multifaceted funds

Polyhedrons are classified and named based on the number of their faces and based on classical Greek; For example, tetrahedron means a polyhedron with four faces, pentahedron means a polyhedron with five faces, hexahedron means a polyhedron with six faces, and so on.

### interior angle of polyhedron

Polyhedra are made from regular polygons. Like polygons, polyhedra have internal and external angles.

The internal angle of polyhedra is obtained based on the number of faces and sides of the vertex.

For example, a tetrahedron has four equilateral triangles, and the interior angle of its face is 60 degrees, and the sum of its interior angles is 180 degrees. But a tetrahedron has a total interior angle of 720 degrees.

So the sum of the internal angle and the size of the internal angle are written accordingly.

$180n[(n'-2)]$

${\frac {180}{n'}}n[(n'-2)]$

### Shape and corners

For each vertex, a corner shape can be defined, which defines the shape of the polyhedron around the vertex. The exact definitions are variable, but a corner shape can be defined as a shape that is created by cutting the vertex of a polyhedron. If the polygon resulting from this process is regular, the vertex is considered regular.

#### vertex symbol

A vertex symbol or vertex configuration is a shorthand symbol for representing the corner shape of a polyhedron or tiling as a sequence of faces around a vertex. For uniform polyhedra there is only one type of corner shape and thus the vertex configuration completely defines the polyhedra.

The vertex symbol is represented as a sequence of numbers that represent the number of sides of the faces around the vertex. The notation "a.b.c" describes a vertex that has 3 faces around it, with sides a, b, and c.

#### Face configuration

Uniform binomials that are face symmetric can be represented by the same abbreviations as the vertex configuration, which is called the face configuration. These symbols are indicated by a V for difference. This symbol is defined as a consecutive count of the number of faces that are located at the vertices around the face. For example, the dodecahedron face configuration is V3,4,3,4 or 2(3,4)V.

### volume

Polyhedral solids have a specific value called volume, which measures the amount of space they occupy. Simple families of polyhedra may have simple formulas for their volumes. For example, the volume of pyramids, prisms, and parallelograms can be easily expressed in terms of side lengths or other specifications.

The volume of more complex polyhedra may not have simple formulas. By dividing the polyhedron into smaller parts, the volume of these polyhedrons is calculated. For example, the volume of a regular polyhedron can be calculated by dividing it into equal pyramids, such that each pyramid has one face of the polyhedron as its base and the center of the polyhedron as its vertex.

In general, it can be derived from the divergence theorem that the volume of a polyhedral solid is given by:

${\frac {1}{3}}\left|\sum _{F}(Q_{F}\cdot N_{F})\operatorname {area} (F)\right|,$

where the sum over the faces of F is the polyhedron, is an arbitrary point on the face of F, is the unit vector perpendicular to F to the outside of the polyhedron, and is the point of multiplication of the inner product.

### area

The area of regular polyhedra has a surface area. The faces of regular polyhedra are regular polygons. Fixed polyhedra such as prisms, pyramids, and parallelograms have constant areas. It is regular, it is obtained based on the sum of the polygonal area and the lateral area of the prism (polygonal area x height).

Area of the polyhedron:$n[{\frac {1}{4}}n'a^{2}cot({\frac {\pi }{n'}})]$ .

n here is the number of faces and 'n is the number of sides of the polygon, here the number π is in radians

### Ashlefly symbol

Schleffle notation is a notation for regular polytopes, including regular polytopes.

Regular polyhedra are denoted as {p,q}, where q is the Schleffle symbol of the corners and p is the Schleffle symbol of the polygon of each face.

The Schleffle symbol is a convex regular polygon of the form {p}, where p is the number of sides. The Schleffle symbol of a regular concave (star) polygon is in the form {p/q}, where p is the number of vertices and q is the number of sides between two vertices in connecting the vertices of the regular convex polygon to make it.

Polyhedra whose Schleffle symbols are similar are duals of each other.

# Theory of sets

Set theory is a fundamental branch of mathematics that deals with the study of sets, i.e. combinations of objects. All mathematics as commonly taught is formulated in the language of set theory and is built upon the axioms of set theory. Most mathematical objects that are treated in subareas such as algebra, analysis, geometry, stochastics or topology can be defined as sets. Measured against this, set theory is a fairly young science; only after overcoming the fundamentals crisis in mathematics in the early 20th century was set theory able to take its central and fundamental place in mathematics today.

# Logic (the study of reasoning)

Reasoning is the lawful combination of known proposition(s) to arrive at new proposition(s). In reasoning, the mind makes a connection between several theorems so that a conclusion is born from their connection and thus a doubtful and ambiguous relationship becomes a certain relationship, in other words, to sum up affirmations (news sentences are said to deny or affirm something else) It has. For example: Mercury is a metal. Man is not an animal.) Another argument is said for affirmative proof.

## Types of reasoning

Arguments are divided into two main categories:

1. Inductive reasoning
2. Deductive reasoning

### Inductive reasoning

Since Aristotle, induction (Latin inducere 'cause', 'cause', 'introduce') means the abstract conclusion from observed phenomena to a more general knowledge, such as a general concept or a law of nature.

The expression is used as an antonym for deduction. A deduction concludes from given assumptions to a special case, whereas induction is the opposite way. Since the middle of the 20th century in particular, how this is to be determined has been the subject of controversial debate, as has the question of whether induction and deduction correspond to actual cognitive processes in everyday life or in science, or whether they are artefacts of philosophy.

David Hume took the position that there can be no induction in the sense of an inference about general and necessary laws that is compelling and experiential. In the 20th century, theorists such as Hans Reichenbach and Rudolf Carnap attempted to develop formally exact theories of inductive reasoning. Karl Popper vehemently tried to show that induction is an illusion that in reality only deduction is used and that it is sufficient. Up until his death he made the controversial claim that he had actually and definitively solved the problem of induction with his deductive methodical approach.

Various attempts have been made throughout the 20th century to defend the notion of induction against criticism from, for example, Hume, Nelson Goodman, and Popper. In this context, various theories of inductive reasoning and more general inductive methods were developed (especially with recourse to Bayesian theory of probability) and empirical studies were carried out. Questions related to the concept of induction today fall into sub-areas of philosophy of mind, philosophy of science, logic, epistemology, theory of rationality, argumentation and decision-making, psychology, cognitive science and artificial intelligence research.

From a logical point of view, the mathematical procedure of complete induction is not an inductive conclusion; on the contrary, it is a deductive method of proof.

### Deductive reasoning

Deduction (Latin deductio, derivation, continuation, derivation), also deductive method or deductive conclusion, is the process of drawing logically compelling conclusions. A conclusion is compellingly or deductively valid if its conclusion follows logically from the premises. The truth of the premises (also assumptions, prerequisites) must therefore be sufficient for the truth of what is deductively concluded (the conclusion).

In Aristotelian logic, deduction was traditionally only used as a "conclusion from the general to the particular", i. H. the inheritance of the properties that all elements of a group share, to real subsets and individual elements. Aristotle contrasted this with induction (obtaining general statements from the consideration of several individual cases) and abduction or apagogue (the determination that certain individual cases fall under a given or yet to be discovered general rule). With the development of modern logic, however, an understanding of deduction as a formal relationship between logical statements has been established. A conclusion is valid if there is no possible case in which the premises can be true and the conclusion false. The validity of the derivation according to a clear rule of inference makes up the essence of deduction in the modern understanding.

Deductive reasoning is studied in logic, psychology, and the cognitive sciences.Some theorists emphasize the difference between these areas in their definition. According to this view, psychology studies deductive reasoning as an empirical mental process, i. H. she examines what happens when people draw conclusions. But the descriptive question of how reasoning actually works is different from the normative question of how it should work or what constitutes correct deductive reasoning, which is studied by logic.This is sometimes expressed by the fact that logic, strictly speaking, does not study deductive reasoning, but rather the deductive relationship between the premises and a conclusion called a logical consequence. However, this distinction is not always strictly adhered to in the scientific literature. An important aspect of this difference is that logic does not care whether an argument's conclusion makes sense. From the premise "the printer has ink" one can draw the unhelpful conclusion "the printer has ink and the printer has ink and the printer has ink", which is of little psychological relevance. Instead, actual thinkers usually try to remove redundant or irrelevant information and make the relevant information clearer. The psychological study of deductive reasoning also addresses how good people are at making deductive reasoning and the factors that determine their performance. Deductive reasoning can be found both in natural language and in formal logical systems such as propositional logic.

Dealing with deduction plays a central role in logic and the philosophy of science of the 20th century.

# Number Theory

Number theory is one of the branches of pure mathematics, which is mainly devoted to the investigation and study of functions of integers, arithmetic functions, and functions of natural numbers. German mathematician Carl Friedrich Gauss (1855-1777) said: "Mathematics is the queen of sciences, and number theory is the queen of mathematics."

Students of number theory, prime numbers, as well as the characteristics of numbers from numbers such as integers (such as rational numbers), or defining it as generalizing one number to another number to use for some aspects of numbers as convenient working and completion with it For example, they study integers to use coordinates and algebraic integers, etc.

First of all, integers are used in other places where it is necessary to calculate area and volume, use as calculation of integration and Fourier series expansions, etc. and it can be said that it is used by everyone and cannot be seen anywhere without its application. Secondly, he considered integers as a solution for equations, for example Diophantine geometry equations. Questions in number theory are often studied through the study of analytic objects and complex states (for example, Riemannian zeta functions) in which the properties of natural numbers and Arithmetic and integers and prime numbers are coded (analytical number theory) in the best way for Bai to understand. He also used real numbers as a study about rational numbers, for example Diophantine's approximation.

# Graph Theory

Graph theory or topology graph theory is a branch of mathematics and a field of discrete mathematics. Graph theory refers to networks of points that are connected and ordered and connected by lines. Graph applications in chemistry for formulation and form Classification of molecules and atoms, research and investigation in operations for tracking and clues in a research and computer science study have been used a lot to encode and determine the graph, and the graph has become a significant field in mathematics and discrete mathematics.

The history of graph theory goes back to 1735 in a special, principled and official way, which was discovered by a Swiss mathematician named Leonard Euler. It was a mathematical puzzle to find a path on each of the seven bridges that cross a forked river and pass by an island. Euler created an argument by drawing graph lines that there is no path at all. Proof This theorem was in the form of Fyric arrangement in bridges, but basically he proved the first theorem in graph theory.

The term graph means that the lines are connected to each other and have a graph scope, but it does not refer to data graphs such as linear, bar, circle. In general, graphs can be divided into two categories: graphs and Divided data. A graph refers to a set of vertices (points or nodes that are connected) and lines (lines that are connected).

# Classification of data

Data classification is a type of quantity in statistics that calculates data of more than 10 and creates sets of multiple categories.

This classification even calculates the average of more than 10 data

This topic in applied mathematics in the subject of statistics is very good for categories of height in medium, short, tall, profit and loss in economy and tax, etc.

## Definitions

### Range of Changes

The difference between two numbers, one of which is the largest data and the other is the smallest data, is called the range of changes.

### categories

The number of specific categories in the categories, which is considered a type of variable, is said to divide the difference between the largest and the smallest data by its number. This category shows which data should be categorized in its range.

### Abundance

The result of collecting one category within the category is called that there are several data in the collection.

### handle center

The center of the category means the average of two categories, which is equal to their sum divided by two, which is used for the average of the data.

# Law of total probability

The law of total probability, which is also known as Bayes' law of probability, is a rule of law in the field of statistics and probability that divides probability calculations into a separate, detailed and advanced part, that is, for example, you have no relative knowledge of the probability of the occurrence of an expression B, and for Finding the probability of an event B that we know, we can use to find the probability of the event B.

## Multivariate Law of Total Probability

According to this pattern B1,..., Bn and an event called set A, the probability of event A can be calculated as a quantitative or weighted average in which the pattern B1,..., Bn plays a role to find the calculation of its occurrence. According to each occurrence in the partition pattern with the probability of the occurrence of partitioning in terms of quantity, this formula can be used:

$\mathrm {P} (\mathrm {A} )=\mathrm {P} (\mathrm {A} |\mathrm {B} _{1})\mathrm {P} (\mathrm {B} _{1})+.....+\mathrm {P} (\mathrm {A} |\mathrm {B} _{n})\mathrm {P} (\mathrm {B} _{n})$

The same idea can be applied and calculated for random vectors. In this way, it is possible to obtain a

For the marginal distribution, we pooled other variables from the joint distribution:

$f(\mathrm {X} )\mathrm {X} _{1}=\int \limits _{x_{2}}^{}...\int f\mathrm {X} _{1}\mathrm {X} _{2}(x_{1},x_{2})dx_{2}$

The law of total probability is both an independent event and the probability of evil, and it is a general law for these two topics. From the definition of conditional probability, we know that

## References

1. https://pages.uoregon.edu › Law of Total Probability Given a sequence of mutually exclusive ...
2. http://web.mit.edu ›Handout on multivariate law of total probability and BayesXRule
3. https://www.stat.auckland.ac.nz › ...PDF Chapter 2: Probability

# Real analysis

Real analysis or real-scientific analysis is a branch of mathematics analysis that studies and analyzes the behavior of real numbers, real sequences and series of real numbers and real functions.

Convergences, limits, continuity, smoothness, differentiability and integration are the characteristics of real-valued sequences and functions that are checked by real analysis. One of the applications of mathematical analysis is in integral and differential calculus and topology, of course. Its applications are analyzed and acted on based on real ideas. In a wide range of these applications and research, real-time analysis has become a vital tool for research and analysis for these ideas.

# References

## Book References

• calculus, James Stewart, published in 2008
• Calculus with Analytic Geometry, Richard A. Silverman, published in 2010
• Algebra and trigonometry, published by the authors of the college 2021.
• Fourier series,Fourier transform andTheir applications in mathematics andPhysics ,Springer Publishing.
• Proofbook, Richard Hammock, Virginia Commonwealth University, 2010
• advanced Modern engineering mathematics,Glyn James,2022