Mathematical Physics

The mathematical toolkit a working physicist actually uses.

About this course

Fourteen lectures assembling the mathematics physics depends on. It opens with vector calculus in index notation, curvilinear coordinates and tensors, then develops complex analysis through analytic functions, contour integration and the residue theorem — including the real integrals it makes tractable. The third module covers series solutions, Sturm-Liouville theory and the special functions, and the last turns to the calculus of variations, Fourier methods, partial differential equations and Green's functions.

What you will be able to do

  • Work fluently in index notation and apply the field operators in curvilinear coordinates
  • Manipulate tensors and their transformation laws
  • Use contour integration and the residue theorem to evaluate real integrals
  • Solve ODEs by series methods and apply Sturm-Liouville theory and the special functions
  • Apply variational methods, Fourier transforms and Green's functions to physical PDEs

What this course covers

  • Linear second-order ODEs and solution structure
  • Ordinary versus regular singular points
  • The power series method at an ordinary point
  • Special functions from physical symmetries
  • The Frobenius method and the indicial equation
  • Frobenius applied to Bessel's equation: constructing J₀
  • The recurrence relation as a function's complete definition
  • Functionals and the variational problem
  • Derivation of the Euler–Lagrange equation
  • The Beltrami identity
  • Euler–Lagrange applied: shortest path and brachistochrone
  • The principle of least action and Newton's second law
  • Variational principles as the foundation of physics
  • Worked example: harmonic oscillator via least action
  • The complex plane and polar representation
  • Complex functions as plane mappings
  • The complex derivative and its directional demand
  • Derivation of the Cauchy-Riemann equations
  • Analytic functions and the Cauchy-Riemann test
  • Harmonic parts and the structure of analytic functions
  • Reconstructing an analytic function from its real part
  • The del operator and its three derivatives
  • Index notation and the Einstein summation convention
  • Kronecker delta and Levi-Civita symbol
  • The epsilon-delta identity and its application
  • Divergence theorem and Stokes' theorem
  • Index notation as the course's foundational principle
  • Motivation for curvilinear coordinates
  • Scale factors and the curvilinear toolkit
  • Cylindrical and spherical coordinate systems
  • Field operators in curvilinear coordinates
  • The Laplacian master template
  • Proof that ∇²(1/r) = 0 away from the origin
  • Pitfalls of curvilinear coordinates
  • Fourier Analysis as Projection onto a Complete Basis
  • The Fourier Series
  • The Fourier Transform and Its Inverse
  • Square Wave Decomposition and the Gibbs Phenomenon
  • Key Properties: Differentiation, Convolution, and Parseval
  • The Laplace Transform as a Cousin Tool
  • Worked Transform: Box Function to Sinc
  • Differential operators as infinite-dimensional Hermitian matrices
  • Sturm–Liouville form of differential equations
  • Weighted inner product and self-adjointness of the Sturm–Liouville operator
  • Three spectral consequences: real eigenvalues, orthogonality, completeness
  • Four-line proof of orthogonality from self-adjointness
  • Eigenfunction expansion: worked example with the plucked string
  • Sturm–Liouville theory as the mathematical skeleton of quantum mechanics
  • Recap: analyticity and the Cauchy-Riemann equations
  • Contour integrals: definition and parameterisation
  • Cauchy's integral theorem
  • Path independence and contour deformation
  • Cauchy integral formula and infinite differentiability
  • The key integral: loop of 1/(z−z₀) equals 2πi
  • Laurent series, residues, and preview of the residue theorem
  • Symmetry of the Laplacian summons special functions
  • Legendre polynomials
  • Bessel functions
  • Hermite polynomials and quantum harmonic oscillator
  • Deriving P2 and proving orthogonality to P1
  • Legendre expansion of a function
  • Unifying principle: three families, one Laplacian
  • The Residue: Definition and Meaning
  • Computing Residues: Limit and Derivative Formulas
  • The Residue Theorem
  • Closing Contours to Evaluate Real Integrals
  • Jordan's Lemma and Fourier-Type Integrals
  • Evaluating the Nightmare Integral: cos x/(1+x²)
  • Extensions: Higher-Order Poles and Trigonometric Integrals
  • Abstract vector spaces
  • Inner products, length, and orthogonality
  • Linear operators and matrix representations
  • Eigenvalues and eigenvectors
  • Hermitian operators and the spectral theorem
  • Diagonalizing the coupled-oscillator system
  • The eigenbasis principle as the course's third clause
  • The Three Great Linear PDEs of Physics
  • Separation of Variables: the Core Move
  • Boundary Conditions and Sturm-Liouville Mode Selection
  • General Solution via Superposition and Fourier Fitting
  • Universality of Separation: Geometry Chooses Special Functions
  • Worked Example: Uniform Initial Temperature on a Bar
  • Clause Thirteen: the Synthesis Method
  • Tensors as frame-independent geometric objects
  • The tensor transformation law and tensor rank
  • Contravariant and covariant indices
  • The metric tensor
  • Tensor operations: outer product, contraction, and symmetry decomposition
  • Physical tensors: inertia, stress, and electromagnetic field
  • Diagonalizing a tensor: principal axes as eigenvectors
  • Motivation: sourced vs. homogeneous equations
  • The Dirac delta function and sifting property
  • The Green's function: definition and superposition formula
  • Green's functions and the Fourier convolution theorem
  • Constructing a Green's function by hand
  • The Green's function as universal propagator
  • Course synthesis: the complete mathematical physics toolkit

Syllabus

  1. Vectors, Tensors & Linear Algebra

    1. Vector Calculus & Index Notation
    2. Curvilinear Coordinates & the Field Operators
    3. Linear Algebra for Physics
    4. Tensors & Transformations

  2. Complex Analysis

    1. Analytic Functions & the Cauchy-Riemann Equations
    2. Contour Integration & Cauchy's Theorem
    3. The Residue Theorem & Real Integrals

  3. Differential Equations & Special Functions

    1. ODEs & Series Solutions
    2. Sturm-Liouville Theory & Orthogonal Functions
    3. The Special Functions

  4. Transforms, PDEs & Green's Functions

    1. The Calculus of Variations
    2. Fourier Series & the Fourier Transform
    3. PDEs & Separation of Variables
    4. Green's Functions