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# engineering mathematics ii

## ADVANCED CALCULUS AND NUMERICAL METHOD (18MAT21)

VTU
Subject code
18CS36
Semester
3rd Sem

VTU Engineering Mathematics-2 Advanced calculus and numerical methods aims to prepare the students:

• To familiarize the important tools of vector calculus, ordinary/partial differential equations and power series required to analyze the engineering problems.
• To apply the knowledge of  interpolation/extrapolation and numerical integration technique whenever analytical methods fail or very complicated, to offer solutions.

### Module-1

Vector Calculus:-
Vector Differentiation:
Scalar and vector fields. Gradient, directional derivative; curl and divergence-physical interpretation; solenoidal and irrotational vector fields- Illustrative problems.
Vector Integration: Line integrals, Theorems of Green, Gauss and Stokes (without proof). Applications to work done by a force and flux.

### Module-2

Differential Equations of higher-order:- Second order linear ODE’s with constant coefficients-Inverse differential operators, method of variation of parameters; Cauchy’s and Legendre homogeneous equations. Applications to oscillations of a spring and L-C-R circuits.

### Module-3

Partial Differential Equations(PDE’s):- Formation of PDE’s by elimination of arbitrary constants and functions. Solution of non-homogeneous PDE by direct integration. Homogeneous PDEs involving derivatives with respect to one independent variable only. Solution of Lagrange’s linear PDE. Derivation of one-dimensional heat and wave equations and solutions by the method of separation of variables.

### Module-4

Infinite Series:- Series of positive terms- convergence and divergence. Cauchy’s root test and D’Alembert’s ratio test(without proof)- Illustrative examples.
Power Series solutions:- Series solution of Bessel’s differential equation leading to Jn(x)- Bessel’s function of first kind-orthogonality. Series solution of Legendre’s differential equation leading to Pn(x)-Legendre polynomials. Rodrigue’s formula (without proof), problems.

### Module-5

Numerical Methods: Finite differences. Interpolation/extrapolation using Newton’s forward and backward difference formulae, Newton’s divided difference and Lagrange’s formulae (All formulae without proof). Solution of polynomial and transcendental equations — Newton-Raphson and Regula-Falsi methods( only formulae)- Illustrative examples.
Numerical integration: Simpson’s (1/3)” and (3/8)” rules, Weddle’s rule (without proof) —Problems.