Mathematical Methods II, PHAS0009

  • Course Outline and Lecture Notes (Table of Contents)


    Differential Vector Operators
    • Scalar and Vector Fields
    • Partial Derivatives of Fields
    • Directional Derivative and Gradient
    • The Total Differential of Fields
    • Divergence and Curl of Vector Fields
    • Product Rules
    • 2nd Order Variations of Fields, Laplace Operator


    Multidimensional Integration
    • Line Integrals
    • Area Integrals
    • Polar Coordinates
    • Volume Integrals
    • Cylindrical and Spherical Coordinates
    • Surface Integrals
    • Gauss's Divergence Theorem
    • Stokes's Theorem


    Ordinary Differential
    Equations (ODEs)
    • Examples of ODEs in Physics
    • Fixing the Arbitrary Constants
    • Separable First-Order ODEs
    • Linear First-Order ODEs
    • Perfect Differential Method
    • Second-Order Linear ODEs with Constant Coefficients


    Linear Alegbra
    • Formal Definition of Complex Vector Spaces
    • Linear Dependence and Independence of Vectors
    • Matrices and linear transformations
    • Matrix addition and multiplication
    • Determinants
    • Trace of a matrix
    • Properties of matrices (symmetric, hermitian, unitary)
    • Inverse of a matrix
    • Solving coupled linear equations
    • Eigenvalues and Eigenvectors
    • Matrix Diagonalisation
    • Quadratic Forms

  • Books recommended as core/background reading

    • Mathematical Methods in the Physical Sciences, Mary L. Boas, John Wiley & Sons

    • Mathematical Methods for Physics and Engineering: A Comprehensive Guide,
      K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press