## Mathematics-II for Electrical & Electronics Engineering Stream

### Module-1

Vector Calculus: Introduction to Vector Calculus in EC & EE engineering applications.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.

Applications: Conservation of laws, Electrostatics, Analysis of streamlines and electric potentials.

### Module-2

Vector Space and Linear Transformations: Importance of Vector Space and Linear Transformations in the field of EC & EE engineering applications. Vector spaces: Definition and examples, subspace, linear span, Linearly independent and dependent sets, Basis and dimension. Linear transformations: Definition and examples, Algebra of transformations, Matrix of a linear transformation. Change of coordinates, Rank and nullity of a linear operator, Rank-Nullity theorem.Inner product spaces and orthogonality.

Self-study: Angles and Projections.Rotation, reflection, contraction and expansion.

Applications: Image processing, AI & ML, Graphs and networks, Computer graphics.

### Module-3

Laplace Transform: Importance of Laplace Transform for EC & EE engineering applications. Existence and Uniqueness of Laplace transform (LT), transform of elementary functions, region of convergence. Properties–Linearity, Scaling, t-shift property, s-domain shift, differentiation in the sdomain, division by t, differentiation and integration in the time domain. LT of special functionsperiodic functions (square wave, saw-tooth wave, triangular wave, full & half wave rectifier), Heaviside Unit step function, Unit impulse function.Inverse Laplace Transforms:Definition, properties, evaluation using different methods, convolution theorem (without proof),problems, and applications to solve ordinary differential equations.

Self-Study: Verification of convolution theorem.

Applications: Signals and systems, Control systems, LR, CR & LCR circuits.

### Module-4

Numerical Methods -1: Importance of numerical methods for discrete data in the field of EC & EE engineering applications. Solution of algebraic and transcendental equations: Regula-Falsi method and Newton-Raphson method (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: Trapezoidal, Simpson's (1/3)rd and (3/8)th rules(without proof). Problems.

Self-Study: Bisection method, Lagrange’s inverse Interpolation, Weddle's rule.

Applications: Estimating the approximate roots, extremum values, area, volume, and surface area.

### Module-5

Numerical Methods -2:Introduction to various numerical techniques for handling EC & EE applications. Numerical Solution of Ordinary Differential Equations (ODEs): 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 and Milne’s predictorcorrector formula (No derivations of formulae). Problems.

Applications: Estimating the approximate solutions of ODE for electric circuits.

List of Laboratory experiments (2 hours/week per batch/ batch strength 15)

10 lab sessions + 1 repetition class + 1 Lab Assessment

1 Finding gradient, divergent, curl and their geometrical interpretation and Verification of Green’s theorem

2 Computation of basis and dimension for a vector space and Graphical representation of linear transformation

3 Visualization in time and frequency domain of standard functions

4 Computing inverse Laplace transform of standard functions

5 Laplace transform of convolution of two functions

6 Solution of algebraic and transcendental equations by Regula-Falsi and Newton-Raphson method

7 Interpolation/Extrapolation using Newton’s forward and backward difference formula

8 Computation of area under the curve using Trapezoidal, Simpson’s (1/3)rd and (3/8)th rule

9 Solution of ODE of first order and first degree by Taylor’s series and Modified Euler’s method

10 Solution of ODE of first order and first degree by Runge-Kutta 4th order and Milne’s predictor-corrector method

Suggested software’s: Mathematica/MatLab/Python/Scilab

#### Suggested Learning Resources:

Books (Title of the Book/Name of the author/Name of the publisher/Edition and Year)

Text Books

1. B. S. Grewal: “Higher Engineering Mathematics”, Khanna Publishers, 44thEd., 2021.

2. E. Kreyszig: “Advanced Engineering Mathematics”, John Wiley & Sons, 10thEd., 2018.

Reference Books

1. V. Ramana: “Higher Engineering Mathematics” McGraw-Hill Education, 11th Ed., 2017

2. Srimanta Pal & Subodh C.Bhunia: “Engineering Mathematics” Oxford University Press,3rdEd., 2016.

3. N.P Bali and Manish Goyal: “A Textbook of Engineering Mathematics” LaxmiPublications, 10thEd., 2022.

4. C. Ray Wylie, Louis C. Barrett: “Advanced Engineering Mathematics” McGraw – HillBook Co., New York, 6th Ed., 2017.

5. Gupta C.B, Sing S.R and Mukesh Kumar: “Engineering Mathematic for Semester I and II”, Mc-Graw Hill Education(India) Pvt. Ltd 2015.

6. H.K. Dass and Er. Rajnish Verma: “Higher Engineering Mathematics” S.Chand Publication, 3rd Ed.,2014.

7. James Stewart: “Calculus” Cengage Publications, 7thEd., 2019.

8. David C Lay: “Linear Algebra and its Applications”, Pearson Publishers, 4th Ed., 2018.

9. Gareth Williams: “Linear Algebra with applications”, Jones Bartlett Publishers Inc., 6th Ed., 2017.

10. Gilbert Strang: “Linear Algebra and its Applications”, Cengage Publications, 4th Ed., 2022.

Web links and Video Lectures (e-Resources):

 http://nptel.ac.in/courses.php?disciplineID=111

 http://www.class-central.com/subject/math(MOOCs)