MAT 202 - Mathematical Methods

Syllabus for MAT 202 – Mathematical Methods

This is a compulsory course for B.Tech students in their 3rd semester.
 
Credits (Lec:Tut:Lab)= 3:0:0 (3 lectures weekly)

Prerequisites: Class XII mathematics

Brief Description: The first part is an introduction to multivariable calculus, finishing with the various versions of Stokes' theorem. The second part deals with series of numbers and functions (such as power series and Fourier series) and their applications to solving differential equations. The concepts and techniques covered here are used extensively in the social and natural sciences as well as in engineering to study systems with many dimensions.

Detailed Syllabus:

1. Computer Algebra System (CAS): Equations, solving linear system, function definition, function evaluation, two and three dimensional plots, differentiation, integration, matrices, matrix algebra, simplification of expressions

2. Differential calculus in several variables: Space curves and arc length, functions of several variables, level curves and surfaces, limits and continuity, partial derivatives, tangent planes, chain rule, directional derivatives, gradient, Lagrange multipliers, extreme values and saddle points, 2^nd derivative test

3. Double and triple integrals: Double integrals over rectangles, double integrals over general regions, double integrals in polar coordinates, center of mass, triple integrals, triple integrals in cylindrical coordinates, triple integrals in spherical coordinates, change of variables

4. Vector Integration: Vector fields, line integrals, fundamental theorem, independence of path, Green's theorem, divergence, curl, parametric surfaces, area of a parametric surface, surface integrals, Stokes' theorem, Gauss' divergence theorem.

5. Series and Applications: Limits of sequences, algebra of limits, series, divergence test, comparison and limit comparison tests, integral test, alternating series test, absolute convergence, root & ratio tests, power series, Taylor polynomials and series, power series method for solving ODEs, Legendre's equation, Bessel's equation, orthogonal functions and Sturm-Liouville problem, periodic functions and trigonometric series, Fourier series, half-range expansions, Fourier integral, heat equation

Main References:

• Essential Calculus – Early Transcendentals, by James Stewart. Cengage, India Edition. (Chapters 8 to 13)
• Advanced Engineering Mathematics, Erwin Kreyszig, 9^th edition, Wiley India, 2011.

Supplementary References:

• Advanced Engineering Mathematics, Dennis Zill and Warren Wright, 4^th ed., Jones & Bartlett, 2011.
• Calculus and Analytic Geometry by G B Thomas and R L Finney, 9^th edition, Pearson.
• Basic Multivariable Calculus by J E Marsden, A J Tromba and A Weinstein, 1^st edition, Springer (India), 2011.