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# Engineering mathematics - 2

VTU
Subject code
21MAT21
Semester
2nd Sem

## ADVANCED CALCULUS AND NUMERICAL METHODS (21MAT21)

VTU 2nd Sem Maths -21MAT21 Advanced Calculus And Numerical Methods(M2) Engineering Mathematics Notes.

### Module-1

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 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 Green’s theorem and Stoke’s theorem. Problems.
Self-Study: Volume integral and Gauss divergence theorem.

### Module-3

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

Solution of polynomial and transcendental equations: Regula-Falsi and Newton-Raphson methods (only formulae). Problems.
Finite differences, Interpolation using Newton’s forward and backward difference formulae, Newton’s divided difference formula and Lagrange’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, Lagrange’s inverse Interpolation, Weddle’s rule.

### Module-5

Numerical Solution of Ordinary Differential Equations (ODE’s): Numerical solution of ordinary differential equations of first order and first degree: Taylor’s series method, Modified Euler’s method, Runge-Kutta method of fourth order, Milne’s predictor-corrector formula (No derivations of formulae). Problems.