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Mathematics for physics Students

Calculus of variations

What is the Euler-Lagrange equation

How to calculate the functional derivative

What are the various application of the functional derivative

How to evaluate the first integral

How to use the Lagrangian function

How to use the Noether's theorem

What is a degree of freedom

What is a continuous system

How to obtain natural boundary conditions

What is a Lagrange multiplier

Function spaces

What is a vector space

How to determine the norms of function spaces

What is a Hilbert space

What is an orthonormal set

What is an orthogonal polynomial

What are the various linear operators

What is a distribution

What is a test-function

What is a weak derivative

Linear Ordinary Differential Equations

How to determine existence and uniqueness of solutions to linear ODEs

How to use the Wronskian

How to put an ODE in normal form

How to solve inhomogeneous equations

How to solve differential equations with singular points

What is a concrete operator

Ho to use the adjoint operator

How to determine the completeness of eigenfunctions

Linear Differential Operators

What is a formal operator

Green functions

How to solve inhomogeneous linear equations

How to construct Green functions

How to apply Lagrange's identity

How to expand eigenfunctions

What are the analytic properties of Green functions

How to use the Gelfand-Dikii equation

Partial differential equations

How to classify PDEs

What is a Cauchy data

What is the wave equation

What is the Heat equation

What is the potential theory

Mathematics of real waves

What is the relationship between the speed of propagation and the frequency of a wave

How to make waves

What is a Non-linear wave

What is a soliton

Special functions

What is a curvilinear co-ordinate

What is a spherical harmonic

What is a Bessel function

How to use the Weyl's theorem

Integral equations

What is an integral equation

How to classify integral equations

How to solve an integral equation by using the Fourier Transformation

How to solve singular integral equations

How to solve Wiener-Hopf equations

What are the various series solutions

Vectors and Tensors

How to differentiate covariant and contravariant vectors

What is a tensor

What is a cartesian tensor

Differential calculus on Manifolds

What is a vector field

What is a covector field

How to differentiate tensors

What are the physical applications of

What is a covariant derivative

Integration on manifold

How to integrate p-forms

How to use Stokes' theorem

Differential topology

What is the difference between homeomorphism and diffeomorphism

What is cohomology

What is homology

How to use De Rham's theorem

What is the Poincare duality

How to use characteristics classes

What is the Hodge theory

Groups

What is a group axiom

How to define the representation of a group

What are the physics applications of groups

Lie groups

What are the various matrix groups

What is an invariant vector field

What is a lie algebra

Complex analysis

How to solve Cauchy-Riemann equations

How to perform complex integration

What is a meromorphic function

What is the relationship between analytic functions and topology

Special functions and complex variables

What is the Gamma function

How to solve linear differential equations with coefficients that are meromorphic functions

How to solve ODEs via contour integrals