Undergraduate Mathematics

What Is This Book For?

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 Begin

We expect that the reader have the level usually required of a student starting a university level course that heavily involves 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.

Contents

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 Courses

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

Second Year Courses

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

Third and Fourth Year Courses

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