Mathematics - II

Pre-requisites: 

Single Variable Calculus, Matrix & Determinant, and Vector Algebra (Class 12 NCERT level)

Course Syllabus: 

[Unit 1] - Vector Calculus: Differentiation of vectors, gradient, divergence, curl and their physical meaning. Identities involving gradient, divergence and curl. Line and surface integrals. Green’s, Gauss and Stroke’s theorem and their applications.

[Unit 2] - Linear Algebra: Introduction to linear equations, matrices, Gaussian elimination, LU-Decomposition, inverses. Vector spaces (VS), subspaces, linear independence, span, column space, row space, null space, basis, dimension. Computation of null space, rank, rank-nullity theorem and its applications. Linear transformations, matrix of a linear transformation, change of basis, similarity, determinants. Eigenvalues, eigenvectors, characteristic polynomials, minimal polynomials, Cayley-Hamilton theorem. Algebraic multiplicity, Geometric multiplicity, diagonalization. Inner product VS, Gram-Schmidt process, eigenvalues for various types of matrices.

[Unit 3] - Ordinary Differential Equations: First-order linear equations, Bernoulli’s equations, Exact equations and integrating factor, Higher order linear, differential equations with constant coefficients. Partial Differential Equations: First-order linear PDE, quasi-linear PDE, method of characteristics, Cauchy problem, first-order nonlinear PDE of special type.

Course Outcomes: 

Students will be able to understand

1. Vector-valued functions and, the integration associated with vector-valued functions. 

2. Solution procedure and challenges related to a system of linear equations. 

3. Vector space, linear transformation and importance of eigenvalues & eigenvectors. 

4. Fundamentals of differential equations (ordinary and partial) 

Recommended Books: 

Text Books:

1. B S Grewal, Higher Engineering Mathematics, Khanna Publishers, 44th edition, (2017).

2. G.B. Thomas and R.L. Finney, Calculus and Analytic Geometry, Pearson Education, 9th edition, (2007).

3. R.K. Jain and S.R.K Iyenger, Advanced Engineering Mathematics, Narosa Publishing House, 11th edition, (2011).

4. Kreyszig Erwin, Advanced Engineering Mathematics, John Wiley, 8th edition, (2006).

Reference Books:

1. V.K. Krishnamurthy, V.P. Mainra and J.L. Arora, An Introduction to Linear Algebra, Affiliated East West Press, (1976).

2. Stewart James., Essential Calculus, Thomson Publishers, 6th edition, (2007).

3. G.F. Simmons, Differential Equations (With Applications and Historical Notes), Tata McGraw Hill, (2009).

4. Gilbert Strang, Linear Algebra and Its Applications, Cengage Learning, 4th edition, (2006).

Evaluation Scheme: 

Students will be evaluated on a scale of 100 = (0.75*E1+0.5*E2+0.5*E3) where

E1 - 2 Open Book Exams of 50 Marks each for 2 Hr duration

E2 - 1 Quiz (following level of GATE/IES/UPSC) of 30 Marks for 1 Hr duration

E3 - 1 Poster Presentation on Self-Learnt Topic of 20 Marks for 15 Mins duration

Note: A minimum of 20 marks are required to pass the course. Also, Highest grade (A) will be allotted to students scoring 85+. The grading will be relative-scaled.

Exam/Test Schedule: 

1. First Open-book exam will be held on March 12, 2025.

2. Presentation on self-learning topics 1-4 will be held on February 22, 2025. 

3. Presentation on self-learning topics 5-8 will be held on March 01, 2025.

4. Presentation on self-learning topics 9-10 will be held on March 08, 2025.

5. Presentation on self-learning topics 11-19 will be held on March 30, 2025.

6. Presentation on self-learning topics 11-19 will be held on April 13, 2025.

7. Presentation on self-learning topics 20-21 will be held on April 26-27, 2025.

8. Quiz (pattern of GATE/IES/UPSC) will be held on April 30, 2025.

9. Second Open-book exam will be held on May 12, 2025.