Teaching

Welcome to the Univariate and Multivariate Calculus webpage. You have to visit this page reguarly for any announcement regarding this course.

The Real Number System: Problem Set 01
Sequences and Their Convergence: Problem Set 02
Cauchy Sequences and Subsequence: Problem Set 03
Continuity and Limits: Problem Set 04
Properties of Continuous Functions: Problem Set 05
Differentiability: Problem Set 06
Mean Value Theorem: Problem Set 07
Local Extrema and Points of Inflection: Problem Set 08
Taylor's Theorem Problem Set 09
Series Problem Set 10
Convergence Tests I: Comparison, Limit comparison and Cauchy condensation tests Problem Set 11
Convergence Tests II: Ratio, Root and Leibniz’s tests Problem Set 12
Power Series Problem Set 13
Riemann Integration Problem Set 14
FTC, Riemann sum and Improper Integral Problem Set 15
Functions of Several Variables Problem Set 16
Local Extrema and Saddle Points Problem Set 17
Double Integrals Problem Set 18
Triple Integrals Problem Set 19

Convex Analysis: convex sets, convex functions, calculus of convex functions.
Optimality of Convex Programs: 1st order necessary and sufficient conditions, KKT conditions.
Duality: Lagrange and conic duality.
Linear and Quadratic Programs.
Conic Programs: QCQPs, SOCPs, SDPs.
Smooth Problems: gradient descent, Nesterov's accelerated method, Newton's methods.
Non-smooth Problems: sub-gradient descent.
Special topics: active set and cutting planes methods, proximal point method.

S. Boyd and L.Vandenberghe, Convex Optimization. Cambridge University Press, 2004.

R. T. Rockafellar. Convex Analysis. Princeton University Press, 1996.
G. C. Calafiore and L. El Ghaoui, Optimization Models, Cambridge University Press, 2014.

Laplace Transforms: Definition and properties, Sufficient condition of Existence, Transforms of derivatives and integrals, Derivatives and integrals of transforms, Inverse Laplace Transforms, Exponential shifts, Convolutions, Applications: Differential and Integral Equations.
Fourier Series: Periodic functions, fundamental period, Trigonometric series, Fourier series, Bessel's inequality, Orthonormal and orthogonal set, Euler formulas, Functions with arbitrary periods, Even and odd functions , Half range expansions, Fourier coefficients without integration, Approximation by trigonometric polynomials, Application to differential equation.
Fourier Transforms: Fourier integral theorem, Sine and Cosine Integrals, Inverse Transforms, Transforms of Elementary Functions, Properties, Convolution, Parsevals relation, Transform of Dirac Delta Function, Multiple Fourier transform, Finite Fourier transform.
Z Transforms: Z-transforms, properties, Inverse Z- transforms, relationship with Fourier transforms.
Complex Analysis: Complex numbers, Modulus, Argument, Curves and regions in complex plane, Functions, Limits, Derivatives, Analytic functions, Cauchy-Riemann equations, Complex exponential logarithms and trigonometric function, General powers, Line integrals, Cauchy's theorem, Cauchys integral theorem, Cauchys integral formula, Taylor and Laurent series , Zeros and singularities, Residues, Residues theorem, Evaluation of real improper integrals.

M. Braun, Differential Equations and Their Applications, Springer-Verlag, New York.
W. Trench, Elementary Differential Equations.
J. Schiff, The Laplace Transform: Theory and Applications, Springer.
J. Brown and R. Churchill, Complex Variables and Application, McGraw-Hill.
G. F. Simmons, Differential Equations, Tata Mcgraw Hill.
R. Jain and S. Iyenger, Advanced Engineering Mathematics, Narosa.