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Mathematics Induction

This is a self-guided review of mathematics material that will be useful for the core HetSys training modules, and is designed to be completed at your own pace.

Objectives

By the end of this session you should be able to:
  1. manipulate and carry out operations on vectors
  2. manipulate and carry out operations on matrices
  3. demonstrate a familiarity with basic linear algebra
  4. differentiate and integrate functions
  5. solve first-order and higher-order ordinary differential equations
  6. solve simple cases of some partial differential equations
  7. understand basic ideas of statistics and probability

The material presented here is not intended to be rigorous, but may rather be considered as a pragmatic quick-start guide to the standard of mathematics expected of students enrolled on this course. The Maths for Chemists booklet developed by Birmingham and Leeds also provides a useful refresher (whether or not you consider yourself a chemist!). This list is incomplete, so if while studying you feel you need to brush up on other areas of mathematics, follow the Supporting Material link at the end of the list below.

  1. Vectors
    1. Introduction to vector algebra
    2. Vectors in component form
    3. Vector products
  2. Matrices
    1. Introduction to matrices
    2. Matrix algebra
    3. Matrix inversion
    4. Eigenvalues and eigenvectors
    5. Linear independence
    6. Similarity transforms
    7. Matrix diagonalisation
  3. Linear vector spaces
    1. Definitions and useful properties
    2. Matrix operators and change of basis
  4. Differentiation and integration
    1. Functions and limits
    2. Elementary differentiation
    3. Differentiation of products, quotients and logarithms
    4. Elementary integration
    5. Definite integrals
    6. Integration by parts
    7. Partial differentiation
  5. Ordinary differential equations
    1. Introduction
    2. First-order homogenous ODEs
    3. First-order integrating factor method
    4. Second-order homogenous ODEs (complementary function)
    5. Second-order inhomogeneous ODEs (particular integral)
    6. Second-order variation of parameters
    7. Second-order power series method
  6. Partial differential equations
    1. Fourier series method
    2. Solving PDEs
    3. Weak formulation of PDEs
  7. Probability and statistics
    1. Averages - mean, mode and median
    2. Measures of dispersion/scatter
    3. Definitions and rules of probability
    4. Random variables
  8. Supporting material on other topics can be found from the "Just the Maths" homepage

There is a self-assessment test available to gauge your progress. If you feel sufficiently up-to-date on the maths outlined above, feel free to complete the test now! Otherwise, follow the links to work through each section, and then complete the test.

Acknowledgement: Much of this online material is taken from "Just the Maths", devised by A. J. Hobson , Coventry University. The linear vector spaces and some of the ODE and PDE notes were written by Tiffany Walsh, and the weak form PDE notes were written by Tim Sullivan.