The Department presently offers 10 foundational units of 1 credit each on various topics from basic calculus and linear algebra to differential eqns and probability to the entire community of engineering students of the Institute. These courses have been designed and scheduled in consultation with the different engineering streams so that, as far as possible, a student has the requisite math skills to enrol in any course of his/her choice offered during a particular semester.
Course Name & Syllabus
Elements of Basic Calculus I
Sequences and Series : Limit of a sequence, monotone and Cauchy sequences and properties of convergent sequences, examples. Infinite series, positive series, tests for convergence and divergence, integral test, alternating series, Leibnitz test.
Differential Calculus : Continuity and differentiability of a function of single variable, statement of Rolle’s Theorem, Lagrange’s mean value theorem and applications.
Elements of Basic Calculus II
Integral Calculus : Definite Integrals as a limit of sums, Applications of integration to area, volume,
surface area, Improper integrals.
Functions of several variables : Continuity and differentiability,
mixed partial derivatives, local maxima and minima for function of two variables, Lagrange
Functional Series : Pointwise and uniform convergence, basic aspects of Power series,
Double and Triple Integrals : Calculations, Areas, Volumes, change of variables, Applications.
Integrals of Vector Functions: Line integrals, Green’s formula, path independence, Surface integral:
definition, evaluation, Stoke’s formula, Gauss-Ostrogradsky divergence theorem.
Matrices, Linear equations and solvability, Vector spaces,
Basis and dimension, Linear transforms, Similarity of matrices, Rank-Nullity theorem and its applications.
Eigen values and eigen vectors. Cayley-Hamilton theorem and diagonalization, Inner-product spaces, Gram-Schmidt process.
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’s of special type.
Introduction to Probability
Sample space and events, definitions of probability, properties of probability, conditional
Random variables : Distribution functions, discrete and continuous random variables,
moments of random variables, conditional expectation, Chebyshev inequality, functions of random
Special Distributions : Bernoulli, Binomial, Geometric, Pascal, Poisson, Exponential,
Uniform, Normal distributions, Limit Theorems: Law of large numbers
Laplace and Inverse Laplace transform, linearity, Laplace transforms of Derivatives and Integrals,
partial fractions, unit step function, shifting on the t-axis, periodic functions, applications of Laplace
transform for solving differential equations. Fourier integral, Fourier Sine and Cosine transform,
convolution, applications of Fourier transform for solving differential equations.
Complex Functions limits, Continuity, Differentiability, analytic functions, Cauchy -Riemann
equations, Laplace equations, Harmonic functions, conformal mapping, Cauchy integral theorem,
Cauchy integral formula, derivations of an analytic function,Power series, Taylor series, Laurent
series, zeros, singularities, residues, evaluation of real integrals.
Introduction to Statistics
Random sampling, Estimation of parameters, Confidence Intervals, Testing of Hypothesis,
Goodness of fit, Nonparametric tests, Correlation Analysis.