Syllabus

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, Taylor’s and Maclaurin’s series expansion with remainders, Indeterminate forms Limit, Continuity and Differentiability of functions of several variables, Partial Differentiation, Total Differentiation, Change of variables – Jacobians, Maxima and minima of functions of two and three variables without constraints and with constraints (Lagrange’s method of Multipliers).

Integral Calculus: Riemann integration, Introduction to improper integrals, Beta and Gamma integrals, Differentiation under integral sign Double and triple integrals.

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 plane, Gauss divergence theorem, Stokes’ 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 epsilon method (for sequences as well as for functions of one/several variables).

To introduce the idea behind the 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 and results related to their convergence.

CO2: Students will be able to test the convergence of series of real numbers. They will also get an idea of power series and it’s radius of convergence.

CO3: Students will learn the epsilon-delta approach for limit, continuity and differentiability along with the difference between differential of functions of one variable and several variables.

CO4: 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.

CO5: Students will be able to test the Riemann integrability of elementary functions, to calculate double/triple integrals and change the order of integration, and to test the convergence of improper integrals.

CO6: They will learn about differential operators and vector integration, their properties and applications.

Thomas et. al, Thomas Calculus, Pearson

E. Kreyszig, Advanced Engineering Mathematics, John Wiley and Sons

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