MTH 102: Mathematics
II
Pre-requisite: MTH 101
Credits: 3-1-0-0 [11]
Instructor-In-Charge
& Instructor
for Linear Algebra :
Santosha Kumar Pattanayak, FB 502, email: santosha@iitk.ac.in,
Ph.: 6402, Webpage:
http://home.iitk.ac.in/~santosha/
Instructor
for ODE : Debasis Sen, FB 501,
email: debasis@iitk.ac.in, Ph.: 6401. Webpage:
There
will be 40 Lectures all total : 20 Lectures for Linear Algebra
and 20 for ODE. Each lecture will be of 50 minutes. There will
be tutorial sessions every week and those will be taken by the
assigned tutors. Office hours will be provided by the tutors.
Students can clear their doubts during the office hours or
sending an email to the respective tutor. Feed backs and
suggestions are always welcome and can be communicated by
sending an email to santosha@iitk.ac.in.
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 R^n, 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).
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.
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.
Reference
materials:
(i) E. Kreyzig, Advanced Engineering
Mathematics,
(ii) Lecture Notes by Prof. P. Shunmugraj,
(iii) Lecture
Notes by Prof. Abhijit Pal,
(iv) Lecture Notes by Prof. Arbind Lal,
(v) Lecture Notes by Prof.
S.Ghorai,
(vi) G.
Strang: Linear Algebra, Introduction to linear algebra, 41
Edition, Wellesley Cambridge Press,
(vi) G. F. Simmons: Ordinary Differential Equations,
Differential equations with applications and historical notes,
2nd Edition.
(vii) K. Hoffman, and R. Kunze. Linear Algebra. Prentice-Hall
Inc., 1961.