Syllabus

Module 1 (16 hours)
Differential Calculus: Real number system, Completeness axiom, Sequences (monotone, bounded, Cauchy) and their convergence Series of real numbers, Tests for convergence of Series Limit, Continuity and Differentiability of functions of one variable, Rolle’s Theorem, Mean value theorems, Limit, Continuity and Differentiability of functions of several variables, Partial Differentiation, Total Differentiation, Change of Variables – Jacobians.

Module 2 (10 hours)
Integral Calculus: Riemann integration, Introduction to improper integrals, Beta and Gamma integrals, Differentiation under the integral sign Double and triple integrals.

Module 3 (10 hours)
Vector Calculus: Scalar and Vector fields, Vector differentiation, Gradient of scalar field, Directional derivative, Divergence and Curl of a vector field, Laplacian operator. Line, surface, and volume integrals, Green’s theorem in the plane, Gauss divergence theorem, and Stokes’s theorem.

To introduce the properties of real numbers and give an idea of proofs in real analysis.

To understand the concepts involved in limit via the epsilon method (for sequences as well as for functions of one/several variables).

To introduce the idea behind Riemann integration through upper/lower Riemann sums.

To introduce the concepts of vector differentiation and integration.

CO1: Students will learn about different types of sequences, series, and results related to their convergence.

CO2: Students will learn the epsilon-delta approach for limit, continuity, and differentiability along with the difference between the differential functions of one variable and several variables.

CO3: Students will learn about the applications of basic theorems on continuity and differentiability, and the method of determining maxima/minima of functions of several variables.

CO4: Students will be able to test the Riemann integrability of elementary functions, calculate double/triple integrals change the order of integration, and test the convergence of improper integrals.

CO5: They will learn about differential operators and vector integration, their properties, and applications.

Maurice D. Weir, Joel Hass & Christopher Heil, Thomas’ Calculus Early Transcendentals, Thirteenth Edition, Pearson Education, Inc.

T. M. Apostol, Calculus, Volume I and II, John Wiley & Sons, Inc.

Robert G. Bartle & Donald R. Sherbert, Introduction to Real Analysis, John Wiley & Sons, Inc.

ERWIN KREYSZIG, ADVANCED ENGINEERING MATHEMATICS, John Wiley & Sons, Inc.