Course Name: Calculus (single and multivariable)

Course instructor: Makara Tha

Single Variable Calculus

Unit I: Limits and functions

Introduction to functions, Limits of functions (ε - δ and sequential approach), Algebra of limits, Squeeze theorem, One-sided limits, Infinite limits and limits at infinity; Continuous functions and their properties on closed and bounded intervals.

*Lecture Note: found here

Unit II:  Differentiability  and its application

Differentiability of a real-valued function, Algebra of differentiable functions, Chain rule,

Relative extrema, Interior extremum theorem, Rolle’s theorem, Mean-value theorem, and its

applications, Intermediate value theorem for derivatives, Higher order derivatives and calculation of the nth derivative, Leibniz’s theorem, Taylor’s theorem, Taylor’s series expansions. Indeterminate forms, L’Hôpital’s rule; Concavity and inflexion points; Singular points, Asymptotes, Tracing graphs of rational functions and polar equations.

*Lecture Note: found here

*Status: Keeping updated

Unit III: Integral and its application

Definition of integral, the definite integral, the fundamental theorem of calculus, the indefinite integral, the Net Change theorem, the substitution rule, the technique of integration, area between curves, volumes, arc length, area of a surface of revolution.

Multivariate Calculus

Unit V: Calculus of functions of several variables

Basic concepts, Limits and continuity, Partial derivatives, Tangent planes, Total differential, Differentiability, Chain rules, Directional derivatives and the gradient, Extrema of functions of two variables, Method of Lagrange multipliers with one constraint.

Unit VI: Multiple Integrals

Double integration over rectangular and nonrectangular regions, Double integrals in polar co-ordinates, Triple integrals over a parallelopiped and solid regions, Volume by triple integrals, Triple integration in cylindrical and spherical coordinates, Change of variables in double and triple integrals.

Unit VII: Vector Calculus

Vector field, Divergence and curl, Line integrals and applications to mass and work, Fundamental theorem for line integrals, Conservative vector fields, Green's theorem, Area as a line integral, Surface integrals, Stokes' theorem, Gauss divergence theorem.

References:

[1] James Stewart. Calculus, 8th edition, Cengage Learning, 2016.

[2] Marsden, Tromba & Weinstein. Basic Multivariable Calculus, Springer, 2004.

*Note: Each unit shall be done within three weeks

Assignments

Assignment 01: view here 

*Assign date: May 01, 2026

*Deadline: May 20, 2025 [submit to email loop as PDF format]

*Status: completed.