Table of contents - Mathematics

This is a table of contents for my notes on mathematics. Topics are mainly those used in physics. Posts on complex analysis have been moved to their own dedicated page.

Basics

• Logarithms - a few properties
• Stirling's approximation for large factorials
• Taylor series
• Vector area

Dirac delta function

• Dirac delta function
• Dirac delta function - Fourier transform
• Derivatives of the delta function

Operators and matrices

• Anti-hermitian operators
• Diagonalization of matrices
• Hermitian operators
• Functions of hermitian operators
• Hermitian conjugate (adjoint) of an operator
• Hermitian operators - common eigenfunctions implies they commute
• Hermitian operators - equivalence of conditions
• Hermitian operators - a few theorems
• Inverses of linear operators
• Linear functionals and adjoint operators
• Linear maps, linear operators and commutators
• Linear operators - change of basis, trace and determinant
• Matrix representation of linear operators
• Normal operators
• Projection operators
• Projection operators - formal treatment
• Spectral theorem for normal operators
• Unitary operators
• Unitary operators - active and passive transformations of an operator

Calculus

• Fourier transforms and Plancherel's theorem
• Functionals and functional derivatives
• Gaussian integrals
• Generalized product rule - Leibniz's formula
• Coordinate transformations - the Jacobian determinant
• Jacobian determinants
• Lagrange multipliers

Vector calculus

• Curl and Stokes's theorem
• Divergence & Gauss's Theorem

Linear algebra

• Eigenvalues and eigenvectors
• Gram-Schmidt orthogonalization - a couple of examples
• Inner products and Hilbert spaces
• Null space, range, injectivity and surjectivity
• Orthonormal basis and orthogonal complement
• Span, linear independence and basis
• Subspaces and direct sums
• Vector spaces

Special functions

• Associated Legendre functions
• Associated Legendre functions - orthogonality
• Hermite differential equation - generating functions
• Hermite polynomials
• Laguerre polynomials - normalization
• Legendre equation - Legendre polynomials
• Legendre polynomials - geometric origin
• Legendre polynomials - orthogonality
• Legendre polynomials - Rodrigues formula and orthogonality
• Self-adjoint differential equations