MTH 102: Mathematics II
Pre-requisite: MTH 101
Credits: 3-1-0-0 [11]
Instructor-In-Charge & Instructor for Linear Algebra : Dr. Abhijit Pal, FB 551, email: abhipal@iitk.ac.in, Ph.: 6405, Webpage: https://sites.google.com/site/abhijitwebpage/
Instructor for ODE : Prof. S.Ghorai, FB 559, email: sghorai@iitk.ac.in, Ph.: 7461. Webpage: http://home.iitk.ac.in/~sghorai/
There will be 40 Lectures : 20 Lectures for Linear Algebra and 20 Lectures for ODE. Each lectures will be of 50 minutes. There will be tutorial each week and will be taken by assigned tutors. Office hours will be provided by tutors. Students can clear their doubts during the office hours or mailing to the respective tutors. Lecture notes are available in the website https://sites.google.com/site/abhijitwebpage/lecture-notes. Class notes will be available in https://sites.google.com/site/abhijitwebpage/class-notes. Feedbacks and suggestions are always welcome and can be communicated by mailing to abhipal@iitk.ac.in or by dropping a note in the my mailbox, beside the washroom of Mathematics department.
Linear Algebra: Matrices, System of linear equations, Gauss elimination method, Elementary matrices, Invertible matrices, Gauss-Jordon method for finding inverse of a matrix, Determinants, Basic properties of determinants. Cofactor expansion, Determinant method for finding inverse of a matrix, Cramer's Rule, Vector space, Subspace, Examples, Linear span, Linear independence and dependence, Basis, Dimension, Extension of a basis of a subspace, Intersection and sum of two subspace, Examples. Quotient Space
Quiz-1
Linear transformation, Kernel and Range of a linear map, Rank-Nullity Theorem. Rank of a matrix, Row and column spaces, Solvability of system of linear equations, some applications Inner product on
Cauchy-Schwartz inequality, Orthogonal basis, Gram-Schmidt orthogonalization process. Orthogonal projection, Orthogonal complement, Projection theorem, Fundamental subspaces and their relations, Applications (Least square solutions and least square fittings). Eigen-values, Eigen-Vectors, Characterization of a diagonalizable matrix. Diagonalization: Example, An application. Diagonalization of a real symmetric matrix. Representation of real linear maps by matrices (optional)
Mid-Semester Examination
Ordinary differential equations: Introduction to DE, Order of DE, First Order ODE F(x,y,y')=0. Concept of solution (general solution, singular solution, implicit solution etc.), Geometrical interpretations (direction fields, isoclines), Separable form, Reduction to separable form, Exact equations, Integrating factors (of the form F(x) and F(y)). Linear equations, Bernoulli equation, orthogonal trajectories. Picard's existence and uniqueness theorem (without proof), Picard's iteration method. Numerical methods: Euler's method, improved Euler's method. Second order linear ODE: fundamental system and general solutions of homogeneous equations, Wronskian, reduction of order. Characteristic equations: real distinct roots, complex roots, repeated roots. Non-homogeneous equations: undetermined coefficients.
Quiz-2
Non-homogeneous equations: variation of parameters. Extension to higher order differential equations, Euler-Cauchy equation. Power series solutions: ordinary points (Legendre equation). Power series solutions: regular singular points (Bessel equation), Frobenius method, indicial equations. Legendre polynomials and properties, Bessel functions and properties, Sturm comparison theorem, Sturm-Liouville boundary value problems, orthogonal functions. Laplace transform: Laplace and inverse Laplace transforms, first shifting theorem, existence, transforms of derivative and integral. Laplace transform: Differentiation and integration of transforms, unit step function, Second shifting theorem. Laplace transform: Convolution and applications, initial value problems.
End-Semester Examination
Reference materials:
1. G. Strang: Linear Algebra, Introduction to linear algebra, 41 Edition, Wellesley Cambridge Press.
2. G. F. Simmons: Ordinary Differential Equations, Differential equations with applications and historical notes,
2nd Edition.
3. "E. Kreyzig", Advanced Engineering Mathematics,
4. Lecture Notes by Prof. P. Shunmugraj,
5. Lecture Notes by Prof. Arbind Lal.