Real Analysis

## Sequences and Series of Real Numbers

convergence of sequences

bounded and monotone sequences

Cauchy sequences

Bolzano-Weierstrass theorem

absolute convergence

tests of convergence for series - comparison test

ratio test, root test

Power series (of one real variable)

radius and interval of convergence

term-wise differentiation and integration of power series.

## Functions of One Real Variable

limit

continuity

intermediate value property

differentiation

Rolle's Theorem, mean value theorem

L'Hospital rule

Taylor's theorem

Taylor's series

maxima and minima

Riemann integration (definite integrals and their properties)

fundamental theorem of calculus.

Multivariable Calculus and Differential Equations

## Functions of Two or Three Real Variables

Limit

continuity

partial derivatives

total derivative

maxima and minima.

## Integral Calculus

double and triple integrals

change of order of integration

calculating surface areas and volumes using double integrals

calculating volumes using triple integrals.

## Differential Equations

Bernoulli's equation

exact differential equations

integrating factors

orthogonal trajectories

homogeneous differential equations

method of separation of variables

linear differential equations of second order with constant coefficients

method of variation of parameters

Cauchy-Euler equation

Linear Algebra and Algebra

## Matrices

systems of linear equations

rank

nullity

rank-nullity theorem

inverse

determinant

eigen values

eigen vectors

## Finite Dimensional Vector Spaces

linear independence of vectors

basis

dimension

linear transformations

matrix representation

range space

null space

rank-nullity theorem

## Groups

cyclic groups

abelian groups

non-abelian groups

permutation groups

normal subgroups

quotient groups

Lagrange's theorem for finite groups

group homomorphisms.