Delhi University DSC-11 Multivariate Calculus: Study Guide

DSC-11: Multivariate Calculus is the Semester IV core paper in the B.Sc. (Hons.) Mathematics programme at the University of Delhi (DU) under the Undergraduate Curriculum Framework (UGCF) 2022. It carries 4 credits with 3 lectures and 1 tutorial a week, and it is the multivariable calculus course that follows DSC-5 Calculus and DSC-8 Riemann Integration. This guide follows the official syllabus from the DU Department of Mathematics.

Course at a glance

The learning objectives are to extend single variable differential and integral calculus to functions of two or more variables, to visualise curves and surfaces in the plane and in space, to integrate functions of two and three variables, and to apply these tools to physics, economics and optimisation. By the end you should understand constrained maximisation and minimisation, the relationships between line, double and triple integrals, and Green's, Stokes' and the Gauss divergence theorems.

What DSC-11 covers

• Unit I, Calculus of functions of several variables (18 hours): basic concepts, limits and continuity, partial derivatives, tangent planes, the total differential, differentiability, chain rules, directional derivatives and the gradient, extrema of functions of two variables, and Lagrange multipliers with one constraint.
• Unit II, Double and triple integrals (15 hours): double integrals over rectangular and non rectangular regions, double integrals in polar coordinates, triple integrals over a parallelepiped and over solid regions, volume by triple integrals, cylindrical and spherical coordinates, and change of variables in double and triple integrals.
• Unit III, Green's, Stokes' and Gauss divergence theorems (12 hours): vector fields, divergence and curl, line integrals with applications to mass and work, the fundamental theorem for line integrals, conservative vector fields, Green's theorem, area as a line integral, surface integrals, Stokes' theorem and the Gauss divergence theorem.

DSC-11 topic map

Example problems at the DSC-11 level

These are original practice problems, not past DU questions.

1. A tangent plane. For z = x^2 + y^2 at (1, 1, 2), z_x = 2 and z_y = 2, so the tangent plane is z = 2 + 2(x - 1) + 2(y - 1).

2. Volume with a triple integral. The volume of the unit ball in spherical coordinates is

3. Area as a line integral. For the ellipse x = a\cos t, y = b\sin t, the area is \frac{1}{2}\oint (x\,dy - y\,dx) = \frac{1}{2}\int_0^{2\pi} ab\,dt = \pi ab.

How to do well in DSC-11

• Draw before you integrate. A quick sketch of the region or solid catches most mistakes in limits of integration.
• Learn the three big theorems as one idea. Green, Stokes and Gauss each relate an integral over a boundary to an integral of a derivative over the inside, just like the Fundamental Theorem.
• Use the tutorial hour to work through computations, since Unit III problems are long and reward a tidy method.
• Practise in short daily sessions on practice and keep the formula sheet close for coordinate changes.