## Calculus of variationsWhat is the Euler-Lagrange equationHow 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 spacesWhat is a vector spaceHow 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 EquationsHow to determine existence and uniqueness of solutions to linear ODEsHow 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 OperatorsWhat is a formal operator |

## Green functionsHow to solve inhomogeneous linear equationsHow 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 equationsHow to classify PDEsWhat is a Cauchy data What is the wave equation What is the Heat equation What is the potential theory |

## Mathematics of real wavesWhat is the relationship between the speed of propagation and the frequency of a waveHow to make waves What is a Non-linear wave What is a soliton |
## Special functionsWhat is a curvilinear co-ordinateWhat is a spherical harmonic What is a Bessel function How to use the Weyl's theorem |

## Integral equationsWhat is an integral equationHow 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 TensorsHow to differentiate covariant and contravariant vectorsWhat is a tensor What is a cartesian tensor |

## Differential calculus on ManifoldsWhat is a vector fieldWhat is a covector field How to differentiate tensors What are the physical applications of What is a covariant derivative |
## Integration on manifoldHow to integrate p-formsHow to use Stokes' theorem |

## Differential topologyWhat is the difference between homeomorphism and diffeomorphismWhat 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 |
## GroupsWhat is a group axiomHow to define the representation of a group What are the physics applications of groups |

## Lie groupsWhat are the various matrix groupsWhat is an invariant vector field What is a lie algebra |
## Complex analysisHow to solve Cauchy-Riemann equationsHow to perform complex integration What is a meromorphic function What is the relationship between analytic functions and topology |

## Special functions and complex variablesWhat is the Gamma functionHow to solve linear differential equations with coefficients that are meromorphic functions How to solve ODEs via contour integrals |