## Linear Algebra

1) Finite dimensional vector spaces over real or complex fields. 2) Linear transformations and their matrix representations  3) Rank and nullity; systems of linear equations,  4) Characteristic polynomial 5) Eigenvalues and eigenvectors 6) Diagonalization, minimal polynomial 7) Cayley-Hamilton Theorem

## Linear algebra

8) Finite dimensional inner product spaces 9) Gram-Schmidt orthonormalization process 10) Symmetric, skew-symmetric  11) Hermitian, skew-Hermitian 12) Normal, orthogonal and unitary matrices  13) Diagonalization by a unitary matrix 14) Jordan canonical form 15) Bilinear and quadratic forms

## Calculus

1) Functions of two or more variables 2) Continuity 3) Directional derivatives 4) Partial derivatives 5) Total derivative 6) Maxima and minima 7) Saddle point 8) Method of Lagrange’s multipliers 9) Double and Triple integrals and their applications to area

## Calculus

10) Volume and surface area 11) Vector Calculus: gradient, divergence and curl, Line integrals and Surface integrals, Green’s theorem, Stokes’ theorem, and Gauss divergence theorem

## Real Analysis

1) Metric spaces 2) Connectedness,  3) Compactness,  4) Completeness 5) Sequences and series of functions 6) Uniform convergence,  7) Ascoli-Arzela theorem  8) Weierstrass approximation theorem 9) Contraction mapping principle  10) Power series

## Real analysis

11) Differentiation of functions of several variables,  12) Inverse and Implicit function theorems 13) Lebesgue measure on the real line 14) Measurable functions 15) Lebesgue integral 16) Fatou’s lemma 17) Monotone convergence theorem 18) Dominated convergence theorem