## 21MAT21

## Module - 1

**Integral Calculus**

Multiple Integrals: Evaluation of double and triple integrals, evaluation of double integrals by change

of order of integration, changing into polar coordinates. Applications to find: Area and Volume by

double integral. Problems.

Beta and Gamma functions: Definitions, properties, relation between Beta and Gamma functions.

Problems.

Self-Study: Center of gravity.

## Module - 2

**Vector Calculus**

Vector Differentiation: Scalar and vector fields. Gradient, directional derivative, curl and

divergence – physical interpretation, solenoidal and irrotational vector fields. Problems.

Vector Integration: Line integrals, Surface integrals. Applications to work done by a force and flux.

Statement of Greenâ€™s theorem and Stokeâ€™s theorem. Problems.

Self-Study: Volume integral and Gauss divergence theorem.

## Module - 3

**Partial Differential Equations (PDE’s)**

Formation of PDE’s by elimination of arbitrary constants and functions. Solution of nonhomogeneous PDE by direct integration. Homogeneous PDEs involving derivative with respect to

one independent variable only. Solution of Lagrange’s linear PDE. Derivation of one-dimensional

heat equation and wave equation.

Self-Study: Solution of one-dimensional heat equation and wave equation by the method of

separation of variables

## Module - 4

**Numerical methods -1**

Solution of polynomial and transcendental equations: Regula-Falsi and Newton-Raphson methods

(only formulae). Problems.

Finite differences, Interpolation using Newtonâ€™s forward and backward difference formulae, Newtonâ€™s

divided difference formula and Lagrangeâ€™s interpolation formula (All formulae without proof).

Problems.

Numerical integration: Simpson’s (1/3)rd and (3/8)th rules(without proof). Problems.

Self-Study: Bisection method, Lagrangeâ€™s inverse Interpolation, Weddle’s rule.

## Module - 5

**Numerical methods -2**

Numerical Solution of Ordinary Differential Equations (ODEâ€™s):

Numerical solution of ordinary differential equations of first order and first degree: Taylorâ€™s series

method, Modified Eulerâ€™s method, Runge-Kutta method of fourth order, Milneâ€™s predictor-corrector

formula (No derivations of formulae). Problems.