Undergraduate Mathematics

Contents

What Is This Book For?Edit

This wikibook is intended as a general overview of undergraduate mathematics. In any one field, it may not have the widest coverage on this wiki but the idea is to present the most useful results with many exercises that are tied in carefully into the rest of the book.

It can be used by readers as a hub to connect their current knowledge to what they want to know, laid out in a traditional textbook style, and for editors as a source to expand out from and create more specific titles.

The project was inspired by the Feynmann Lectures in Physics which feature as recommended reading below, for mathematical physicists. It also owes a debt to the early success of Linear Algebra.

Before We BeginEdit

We expect that the reader have the level usually required of a student starting a university level course that heavily requires mathematics. For example in the UK an A level equivalent is required.

Specifically it would be useful to know skills like this:

  • Be able to perform basic arithmetic with real numbers
  • Be able to find roots of polynomials
  • Know the basic meaning of terms like function and set
  • Be able to roughly sketch simple graphs without plotting large numbers of points
  • Know how to differentiate simple functions with the Sum, Product, Chain and Quotient rules
  • Know how to integrate simple functions by parts and by substitution
  • Be able to use either a scientific hand calculator or an equivalent computer program

If you follow the material in this wikibook, then find yourself stuck not knowing a method we assumed, please try looking for a work in K12 to give you the right skill.

ContentsEdit

Contents

Typically larger courses, such as real analysis, are 20 credit courses in the 360 credit breakdown of an undergraduate degree. So it should not be assumed that all courses are the same in scale. Most courses are assumed to be 10 credit courses but more material may be included to help cover the different course structures internationally.

First Year CoursesEdit

  1. The Meaning and Methods of Proof
    1. Proof by contradiction  (Apr 20, 2014)
    2. Proof by exhaustion  (Apr 22, 2014)
    3. Taking the Contrapositive  (Apr 20, 2014)
    4. Mathematical induction  (Apr 20, 2014)
    5. Proof by infinite descent  (Apr 20, 2014)
  2. Introduction to Newtonian Mechanics  (Apr 20, 2014)
    1. Free body diagram  (Apr 20, 2014)
    2. Projectile motion  (Apr 20, 2014)
  3. Introduction to Statistics  (Apr 20, 2014)
    1. Probability Space  (Apr 21, 2014)
    2. Sample space  (Apr 21, 2014)
    3. Event  (Apr 21, 2014)
    4. Random variable  (Apr 21, 2014)
    5. Distributions  (Apr 21, 2014)
    6. Standard deviation  (Apr 21, 2014)
    7. Variance  (Apr 21, 2014)
    8. Expectation  (Apr 21, 2014)
  4. Multivariate Calculus  (Apr 20, 2014)
    1. Partial derivative  (Apr 21, 2014)
    2. Integration With Respect to One Variable  (Apr 21, 2014)
    3. Path Integrals  (Apr 21, 2014)
    4. Surface Integrals  (Apr 21, 2014)
  5. Introduction to Linear Algebra  (Apr 20, 2014)
    1. Solving Linear Systems  (Apr 21, 2014)
    2. Introduction to the Matrix  (Apr 21, 2014)
    3. Gauss Jordan Elimination  (Apr 21, 2014)
    4. Reduced Row Echelon Form  (Apr 21, 2014)
    5. Rank  (Apr 21, 2014)
    6. The Rank-Nullity Theorem  (Apr 21, 2014)
    7. Vector space  (Apr 21, 2014)
    8. Bases and Dimension  (Apr 21, 2014)
  6. Introduction to Mathematical Programming  (Apr 20, 2014)
    1. The Algorithm  (Apr 21, 2014)
    2. Pick a Language  (Apr 21, 2014)
    3. Automating Processes We've Already Met  (Apr 21, 2014)
    4. Iterative Processes and Chaos  (Apr 21, 2014)
  7. Real Analysis (20 Credits)  (Apr 20, 2014)
    1. Intuition and Continuity  (Apr 20, 2014)
    2. Sequence  (Apr 20, 2014)
    3. Limit of a sequence  (Apr 20, 2014)
    4. Limit of a function  (Apr 20, 2014)
    5. Squeeze theorem  (Apr 20, 2014)
    6. Continuous function  (Apr 20, 2014)
    7. Intermediate value theorem  (Apr 20, 2014)
    8. Differentiable function  (Apr 20, 2014)
    9. Mean value theorem  (Apr 20, 2014)
    10. Rolle's Theorem  (Apr 20, 2014)
    11. Differentiation rules  (Apr 20, 2014)
    12. Integrability  (Apr 20, 2014)
    13. Fundamental theorem of calculus  (Apr 20, 2014)
  8. The History of Mathematics  (Apr 20, 2014)

Second Year CoursesEdit

  1. Introduction to the Theory of Groups
    1. Definition of a Group  (Apr 20, 2014)
    2. Connections between Groups and Symmetry  (Apr 20, 2014)
    3. Group homomorphism  (Apr 20, 2014)
    4. Group Isomorphism  (Apr 20, 2014)
    5. First Isomorphism Theorem  (Apr 20, 2014)
  2. Non-Euclidean Geometry
  3. Ordinary Differential Equations
  4. Discrete Mathematics
    1. Graph Theory or the Theory of Networks  (Apr 20, 2014)
  5. Point-Set Topology
    1. Metric space  (Apr 20, 2014)
    2. Definition of a Topological Space  (Apr 20, 2014)
    3. Open set  (Apr 20, 2014)
    4. Homeomorphism  (Apr 20, 2014)
    5. Connectedness  (Apr 20, 2014)
    6. Compact space  (Apr 20, 2014)
    7. Banach and Hilbert Spaces  (Apr 20, 2014)
  6. Number Theory
    1. Greatest common divisor  (Apr 20, 2014)
    2. Least common multiple  (Apr 20, 2014)
    3. Euclidean algorithm  (Apr 20, 2014)
    4. Extended Euclidean algorithm  (Apr 20, 2014)
    5. Chinese remainder theorem  (Apr 20, 2014)
    6. Pollard's rho algorithm  (Apr 20, 2014)
  7. Mathematical Biology
  8. Mathematical Physics

Third and Fourth Year CoursesEdit

  1. The Group Theory of the Symmetries of Simple Shapes  (Mar 17, 2014)
    1. Cyclic group  (May 23, 2014)
    2. Dihedral group  (May 23, 2014)
    3. Alternating group  (May 23, 2014)
    4. Coset  (May 23, 2014)
    5. Quotient group  (May 23, 2014)
    6. Sylow theorems  (May 23, 2014)
    7. Abelian group  (May 23, 2014)
    8. Burnside's lemma  (May 23, 2014)
  2. Advanced Statistics
  3. Complex Analysis
    1. Line integral  (Apr 21, 2014)
    2. Green's theorem  (Apr 21, 2014)
    3. Stokes' theorem  (Apr 21, 2014)
  4. Algebraic Topology
    1. Paths  (Apr 21, 2014)
    2. Deformation Retraction  (Apr 21, 2014)
    3. Homotopy  (Apr 21, 2014)
    4. The fundamental group  (Apr 21, 2014)
    5. Simplicial complexes  (Apr 21, 2014)
    6. Chain complex  (Apr 21, 2014)
    7. Homology groups  (Apr 21, 2014)
  5. Representation Theory
  6. Cryptography
    1. Caesar Shift  (Apr 21, 2014)
    2. Substitution Ciphers  (Apr 21, 2014)
    3. Frequency Analysis  (Apr 21, 2014)
    4. RSA (cryptosystem)  (Apr 21, 2014)

Further ReadingEdit