VTU Notes

The Complete 18MAT21 | Advanced Calculus and Numerical Methods - Maths Notes

  • 4.9
  • 2018 Scheme | Chemistry Department

Description

Course Learning Objectives:

This course, Advanced Calculus and Numerical Methods (18MAT21) aim to prepare the students:

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


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What You’ll Learn

MODULE-I

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.

(RBT Levels: L1 & L2)


MODULE-II

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 spring and L-C-R circuits.

 (RBT Levels: L1, L2 & L3)


MODULE-III

 Partial Differential Equations(PDE's):- Formation of PDE by elimination of arbitrary constants and functions. Solution of non-homogeneous PDE by direct integration. Homogeneous PDEs involve 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.

(RBT Levels: L1, L2 & L3)


MODULE-IV

 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.

(RBT Levels: L1 & L2)


MODULE-V

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

 (RBT Levels: L1, L2 & L3) 


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