## Units and Topics

## Syllabus

- Differentiation Under Integral Sign with Constant Limits of Integration

- Exact Differential Equations, Equations Reducible to Exact Equations By Integrating Factors

- Complimentary function, particular integrals of differential equation of the type f(D)y = X where X is e
^{ax},sin (ax+b), cos (ax+b), x^{n}, e^{ax}V, xV.

- no formulation of differential equation.

- Definition
- Evaluation of Double Integrals
- Change of order of integration
- Evaluation of double integrals by changing the order of integration and changing to polar form (Examples on change of variables by using Jacobians only).

- SciLab programming is to be taught during lecture hours.

- Definition and evaluation (Cartesian, cylindrical and spherical polar coordinates).

- Different type of operators such as shift, forward, backward difference and their relation.
- Interpolation, Newton interpolation, Newton‐ Cotes formula(with proof).
- Integration by (a) Trapezoidal (b) Simpson’s 1/3rd (c) Simpson’s 3/8th rule (all with proof).
- Scilab programming on (a) (b) (c) (d) is to be taught during lecture hours.

- Exact Differential Equations
- Equations Reducible to Exact Form by Using Integrating Factors
- Linear Differential Equations
- Equation Reducible to Linear Form
- Bernoulli’S Equation
- Simple Application of Differential Equation of First Order and First Degree to Electrical and Mechanical Engineering Problem

1.1 Exact differential Equations, Equations reducible to exact form by using integrating factors.

1.2 Linear differential equations (Review), equation reducible to linear form, Bernoulli’s equation.

1.3: Simple application of differential equation of first order and first degree to electrical and Mechanical Engineering problem (no formulation of differential equation)

- Linear Differential Equation with Constant Coefficient‐ Complementary Function
- Particular Integrals of Differential Equation
- Cauchy’S Homogeneous Linear Differential Equation
- Legendre’S Differential Equation
- Method of Variation of Parameters

2.1. Linear Differential Equation with constant coefficient‐ complementary function,

particular integrals of differential equation of the type f(D)y = X where X is 𝑒^{𝑎𝑥}, sin(ax+b), cos (ax+b), 𝑥^{𝑛}, 𝑒^{𝑎𝑥}V, xV.

2.2. Cauchy’s homogeneous linear differential equation and Legendre’s differential equation, Method of variation of parameters

- Taylor’S Series Method
- Euler’S Method
- Modified Euler Method
- Runga‐Kutta Fourth Order Formula
- Beta and Gamma Functions and Its Properties

3.1. (a)Taylor’s series method (b)Euler’s method (c) Modified Euler method (d) Runga‐Kutta fourth order formula (SciLab programming is to be taught during lecture hours)

3.2 .Beta and Gamma functions and its properties.

- Differentiation Under Integral Sign with Constant Limits of Integration
- Numerical Integration‐ by Trapezoidal
- Numerical Integration‐ by Simpson’S 1/3rd
- Numerical Integration‐ by Simpson’S 3/8th Rule
- Rectification of Plane Curves

4.1. Differentiation under integral sign with constant limits of integration.

4.2. Numerical integration‐ by (a) Trapezoidal (b) Simpson’s 1/3rd (c) Simpson’s 3/8th rule (all with proof). (Scilab programming on (a) (b) (c) (d) is to be taught during lecture hours)

4.3. Rectification of plane curves.

- Double Integration‐Definition
- Evaluation of Double Integrals
- Change the Order of Integration
- Evaluation of Double Integrals by Changing the Order of Integration and Changing to Polar Form

5.1. Double integration‐definition, Evaluation of Double Integrals.

5.2. Change the order of integration, Evaluation of double integrals by changing the order of integration and changing to polar form.

- Triple Integration Definition and Evaluation
- Application of Double Integrals to Compute Area
- Application of Double Integrals to Compute Mass
- Application of Double Integrals to Compute Volume
- Application of Triple Integral to Compute Volume

6.1. Triple integration definition and evaluation (Cartesian, cylindrical and spherical polar coordinates).

6.2. Application of double integrals to compute Area, Mass, Volume. Application of triple integral to compute volume.