Supplementary mathematics/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.

Newton the English scientist

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 $(x,f(x))$. The derivative $f'(x)$ 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.