Calculus, Laplace Transforms and Numerical Techniques (1BMATE201)

Calculus, Laplace Transforms and Numerical Techniques

Course Code 1BMATE201 
CIE Marks 50
Teaching Hours/Week (L:T:P: S) 3:2:0:0 
SEE Marks 50
Total Hours of Pedagogy 40Hours (Theory) + 20Hours Tutorials 
Total Marks 100
Credits 04 Exam 
Hours 3 Hours
Examination type (SEE) Theory

Module-1: Integral Calculus and its applications 

Multiple Integrals: Evaluation of double and triple integrals, change of order of integration,

changing to polar coordinates. Areas and volume using double integration.

Beta and Gamma functions: Definitions, properties, relation between Beta and Gamma functions.

Textbook-1: Chapter-7.1,-7.16.

Module-2: Vector calculus and its applications 

Vector differentiation: Scalar and vector fields, gradient of a scalar field, directional derivatives,

divergence of a vector field, solenoidal vector, curl of a vector field, irrotational vector, physical

interpretation of gradient, divergence and curl and scalar potential.

Vector Integration: Line integrals, Statement of Green’s and Stokes’ theorem without

verification problems.

TextBook-1: Chapter-8.4- 8.14.

Module-3: Numerical Methods-1 

Solution of algebraic and transcendental equations: Regula-Falsi method, and Newton-Raphson

method.

Finite Differences and Interpolation: Forward and backward differences, Interpolation, Newton

forward and backward interpolation formulae, Newton’s divided difference interpolation formula

and Lagrange’s interpolation formula.

Numerical Integration: Trapezoidal rule, Simpson’s 1/3rd rule and Simpson’s 3/8th rule.

Textbook-1: Chapter-28.1, 28.2(2,3), 29.1-29.12, 30.4, 30.6, 30.7, 30.8.

Module-4: Numerical Methods-2

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 method and Adam-Bashforth predictor-corrector method.

Textbook -1: Chapter-32.1-32.10

Module-5: Laplace transforms 

Laplace transforms: Definition and Formulae of Laplace Transforms, Laplace Transforms of

elementary functions. Properties–Linearity, Scaling, shifting property, differentiation in the s

domain, division by t. Laplace Transforms of periodic functions, square wave, saw-tooth wave,

triangular wave, full and half wave rectifier, Heaviside Unit step function.

Inverse Laplace Transforms: Definition, properties, evaluation of Inverse Laplace Transforms

using different methods, and applications to solve ordinary differential equations.

Textbook -1: Chapter-21.1- 21.17

Suggested Learning Resources: 

Textbooks:

1. B.S. Grewal, Higher Engineering Mathematics, Khanna Publishers, 44th Ed., 2021.

2. E. Kreyszig, Advanced Engineering Mathematics, JohnWiley & Sons, 10th Ed.,2018.

3. M.K. Jain, S.R.K. Iyengar and R.K. Jain, Numerical Methods for Scientific and

Engineering Computation, New Age International Publishers, 8th Ed., 2022.

Reference books:

1. B. V. Ramana, Higher Engineering Mathematics, McGraw-HillEducation,11thEd., 2017

2. Srimanta Pal & Subodh C. Bhunia, Engineering Mathematics, Oxford University

Press, 3rd Ed., 2016.

3. N. P. Bali and Manish Goyal, A Text book of Engineering Mathematics, Laxmi

Publications,10thEd.,2022.

4. H. K. Das and Er. Rajnish Verma, Higher Engineering Mathematics, S. Chand

Publication, 3rd Ed., 2014.

5. Steven V. Chapra and Raymond P. Canale, Applied Numerical Methods with Matlab

for Engineers and Scientists, McGraw-Hill, 3rdEd., 2011.

6. Richard L. Burden, Douglas J. Faires and A. M. Burden, Numerical Analysis, 10th Ed.,

2010, Cengage Publishers.

7. S.S. Sastry,“ Introductory Methods of Numerical Analysis”, PHI Learning Private

Limited, 5thEd.,2012.