Undergraduate Mathematics

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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 Begin[edit]

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.

Contents[edit]

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[edit]

  1. The Meaning and Methods of Proof
    1. Proof by contradiction 25% developed  as of Apr 20, 2014 (Apr 20, 2014)
    2. Proof by exhaustion 25% developed  as of Apr 22, 2014 (Apr 22, 2014)
    3. Taking the Contrapositive 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    4. Mathematical induction 25% developed  as of Apr 20, 2014 (Apr 20, 2014)
    5. Proof by infinite descent 25% developed  as of Apr 20, 2014 (Apr 20, 2014)
  2. Introduction to Newtonian Mechanics 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    1. Free body diagram 25% developed  as of Apr 20, 2014 (Apr 20, 2014)
    2. Projectile motion 25% developed  as of Apr 20, 2014 (Apr 20, 2014)
  3. Introduction to Statistics 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    1. Probability Space 25% developed  as of Apr 21, 2014 (Apr 21, 2014)
    2. Sample Space 25% developed  as of Apr 21, 2014 (Apr 21, 2014)
    3. Event 25% developed  as of Apr 21, 2014 (Apr 21, 2014)
    4. Random variable 25% developed  as of Apr 21, 2014 (Apr 21, 2014)
    5. Distributions 25% developed  as of Apr 21, 2014 (Apr 21, 2014)
    6. Standard deviation 25% developed  as of Apr 21, 2014 (Apr 21, 2014)
    7. Variance 25% developed  as of Apr 21, 2014 (Apr 21, 2014)
    8. Expectation 25% developed  as of Apr 21, 2014 (Apr 21, 2014)
  4. Multivariate Calculus 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    1. Partial derivative 25% developed  as of Apr 21, 2014 (Apr 21, 2014)
    2. Integration With Respect to One Variable 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
    3. Path Integrals 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
    4. Surface Integrals 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
  5. Introduction to Linear Algebra 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    1. Solving Linear Systems 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
    2. Introduction to the Matrix 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
    3. Gauss Jordan Elimination 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
    4. Reduced Row Echelon Form 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
    5. Rank 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
    6. The Rank-Nullity Theorem 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
    7. Vector space 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
    8. Bases and Dimension 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
  6. Introduction to Mathematical Programming 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    1. The Algorithm 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
    2. Pick a Language 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
    3. Automating Processes We've Already Met 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
    4. Iterative Processes and Chaos 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
  7. Real Analysis (20 Credits) 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    1. Intuition and Continuity 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    2. Sequence 25% developed  as of Apr 20, 2014 (Apr 20, 2014)
    3. Squeeze theorem 25% developed  as of Apr 20, 2014 (Apr 20, 2014)
    4. Limit of a sequence 25% developed  as of Apr 20, 2014 (Apr 20, 2014)
    5. Continuous function 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    6. Intermediate value theorem 25% developed  as of Apr 20, 2014 (Apr 20, 2014)
    7. Differentiability 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    8. The Mean Value Theorem 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    9. Rolle's Theorem 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    10. Proving the Rules of Differentiability 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    11. Integrability 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    12. Fundamental theorem of calculus 25% developed  as of Apr 20, 2014 (Apr 20, 2014)
  8. The History of Mathematics 0% developed  as of Apr 20, 2014 (Apr 20, 2014)

Second Year Courses[edit]

  1. Introduction to the Theory of Groups 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    1. Definition of a Group 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    2. Connections between Groups and Symmetry 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    3. Group homomorphism 25% developed  as of Apr 20, 2014 (Apr 20, 2014)
    4. Group Isomorphism 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    5. First Isomorphism Theorem 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
  2. Non-Euclidean Geometry 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
  3. Ordinary Differential Equations 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
  4. Discrete Mathematics 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    1. Graph Theory or the Theory of Networks 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
  5. Point-Set Topology 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    1. Metric space 25% developed  as of Apr 20, 2014 (Apr 20, 2014)
    2. Definition of a Topological Space 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    3. Open set 25% developed  as of Apr 20, 2014 (Apr 20, 2014)
    4. Homeomorphism 25% developed  as of Apr 20, 2014 (Apr 20, 2014)
    5. Connectedness 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    6. Compact space 25% developed  as of Apr 20, 2014 (Apr 20, 2014)
    7. Banach and Hilbert Spaces 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
  6. Number Theory 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    1. Greatest common divisor 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    2. Least common multiple 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    3. Euclidean algorithm 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    4. Extended Euclidean algorithm 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    5. Chinese remainder theorem 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    6. Pollard's rho algorithm 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
  7. Mathematical Biology 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
  8. Mathematical Physics 0% developed  as of Apr 20, 2014 (Apr 20, 2014)

Third and Fourth Year Courses[edit]

  1. The Group Theory of the Symmetries of Simple Shapes 0% developed  as of Mar 17, 2014 (Mar 17, 2014)
    1. Cyclic group 25% developed  as of May 23, 2014 (May 23, 2014)
    2. Dihedral group 25% developed  as of May 23, 2014 (May 23, 2014)
    3. Alternating group 25% developed  as of May 23, 2014 (May 23, 2014)
    4. Coset 25% developed  as of May 23, 2014 (May 23, 2014)
    5. Quotient group 25% developed  as of May 23, 2014 (May 23, 2014)
    6. Sylow theorems 25% developed  as of May 23, 2014 (May 23, 2014)
    7. Abelian group 25% developed  as of May 23, 2014 (May 23, 2014)
    8. Burnside's lemma 25% developed  as of May 23, 2014 (May 23, 2014)
  2. Advanced Statistics 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
  3. Complex Analysis 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    1. Line integral 25% developed  as of Apr 21, 2014 (Apr 21, 2014)
    2. Green's theorem 25% developed  as of Apr 21, 2014 (Apr 21, 2014)
    3. Stokes' theorem 25% developed  as of Apr 21, 2014 (Apr 21, 2014)
  4. Algebraic Topology 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    1. Paths 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
    2. Deformation Retraction 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
    3. Homotopy 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
    4. The fundamental group 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
    5. Simplicial complexes 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
    6. Chain complex 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
    7. Homology groups 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
  5. Representation Theory 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
  6. Cryptography 0% developed  as of Apr 20, 2014 (Apr 20, 2014)
    1. Caesar Shift 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
    2. Substitution Ciphers 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
    3. Frequency Analysis 0% developed  as of Apr 21, 2014 (Apr 21, 2014)
    4. RSA (cryptosystem) 0% developed  as of Apr 21, 2014 (Apr 21, 2014)

Further Reading[edit]