## Mathematics-I for Computer Science and Engineering stream

### Module-1

Calculus:Introduction to polar coordinates and curvature relating to Computer Science and Engineering.Polar coordinates, Polar curves, angle between the radius vector and the tangent, angle between two curves. Pedal equations. Curvature and Radius of curvature - Cartesian, Parametric, Polar and Pedal forms. Problems.

Self-study: Center and circle of curvature, evolutes and involutes.

Applications: Computer graphics, Image processing.

### Module-2:

Series Expansion and Multivariable Calculus :Introduction of series expansion and partial differentiation in Computer Science & Engineering applications.Taylor’s and Maclaurin’s series expansion for one variable (Statement only) – problems.Indeterminate forms - L’Hospital’s rule-Problems.Partial differentiation, total derivative - differentiation of composite functions. Jacobian and problems. Maxima and minima for a function of two variables. Problems.

Self-study: Euler’s theorem and problems. Method of Lagrange’s undetermined multipliers with single constraint.

Applications: Series expansion in computer programming, Computing errors and approximations.

### Module-3

Ordinary Differential Equations (ODEs) of First Order:Introduction to first-order ordinary differential equations pertaining to the applications for Computer Science & Engineering.Linear and Bernoulli’s differential equations. Exact and reducible to exact differential equations -Integrating factors on 1/𝑁(𝜕𝑀/𝜕𝑦 −𝜕𝑁/𝜕𝑥) 𝑎𝑛𝑑 1/𝑀(𝜕𝑁/𝜕𝑥 −𝜕𝑀/𝜕𝑦). Orthogonal trajectories, L-R & C-R circuits.Problems.

Non-linear differential equations: Introduction to general and singular solutions, Solvable for p only, Clairaut’s equations,reducible to Clairaut’s equations. Problems.

Self-Study: Applications of ODEs, Solvable for x and y.

Applications of ordinary differential equations: Rate of Growth or Decay, Conduction of heat.

### Module-4

Modular Arithmetic :Introduction of modular arithmetic and its applications in Computer Science and Engineering. Introduction to Congruences, Linear Congruences, The Remainder theorem, Solving Polynomials, Linear Diophantine Equation, System of Linear Congruences, Euler’s Theorem, Wilson Theorem and Fermat’s little theorem. Applications of Congruences-RSA algorithm.

Self-Study: Divisibility, GCD, Properties of Prime Numbers, Fundamental theorem of Arithmetic.

Applications: Cryptography, encoding and decoding, RSA applications in public key encryption.

### Module-5

Linear Algebra:Introduction of linear algebra related to Computer Science &Engineering.Elementary row transformationofa matrix, Rank of a matrix. Consistency and Solution of system of linear equations - Gauss-elimination method, Gauss-Jordan method and approximate solution by Gauss-Seidel method. Eigenvalues and Eigenvectors, Rayleigh’s power method to find the dominant Eigenvalue and Eigenvector.

Self-Study: Solution of system of equations by Gauss-Jacobi iterative method. Inverse of a square matrix by Cayley- Hamilton theorem.

Applications: Boolean matrix, Network Analysis, Markov Analysis, Critical point of a network system. Optimum solution.

#### List of Laboratory experiments

10 lab sessions + 1 repetition class + 1 Lab Assessment

1 2D plots for Cartesian and polar curves

2 Finding angle between polar curves, curvature and radius of curvature of a given curve

3 Finding partial derivatives and Jacobian

4 Applications to Maxima and Minima of two variables

5 Solution of first-order ordinary differential equation and plotting the solution curves

6 Finding GCD using Euclid’s Algorithm

7 Solving linear congruences 𝒂𝒙 ≡ 𝒃(𝒎𝒐𝒅 𝒎)

8 Numerical solution of system of linear equations, test for consistency and graphical representation

9 Solution of system of linear equations using Gauss-Seidel iteration

10 Compute eigenvalues and eigenvectors and find the largest and smallest eigenvalue byRayleigh power method.

#### 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.

3. David M Burton: “Elementary Number Theory” Mc Graw Hill, 7th Ed.,2017.Reference Books

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

5. Srimanta Pal & Subodh C.Bhunia: “Engineering Mathematics” Oxford University Press,3rd Ed., 2016.

6. N.P Bali and Manish Goyal: “A Textbook of Engineering Mathematics” LaxmiPublications, 10th Ed., 2022.

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

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

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

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

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

12. Gareth Williams: “Linear Algebra with Applications”, Jones Bartlett Publishers Inc., 6thEd., 2017.

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

14. William Stallings: “Cryptography and Network Security” Pearson Prentice Hall, 6th Ed.,2013.

15. Kenneth H Rosen: “Discrete Mathematics and its Applications” McGraw-Hill, 8th Ed.2019.

16. Ajay Kumar Chaudhuri: “Introduction to Number Theory”NCBA Publications, 2nd Ed.,2009.

17. Thomas Koshy: “Elementary Number Theory with Applications”Harcourt Academic Press,2nd Ed., 2008.

Web links and Video Lectures (e-Resources):

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

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