# Delhi University DSC-11 Multivariate Calculus: Study Guide

> Study guide for DSC-11 Multivariate Calculus in DU BSc (Hons) Mathematics, UGCF 2022: partial derivatives, multiple integrals and Green, Stokes and Gauss.

Source: https://practicecalculus.com/courses/university-of-delhi-dsc-11
Updated: 2026-11-08

## At a glance

- **Course code:** DSC-11: Multivariate Calculus (Discipline Specific Core 11)
- **Programme:** B.Sc. (Hons.) Mathematics, Undergraduate Curriculum Framework (UGCF) 2022
- **Credits:** 4 credits (3 lecture + 1 tutorial)
- **Semester:** Semester IV
- **Eligibility:** Class XII pass with Mathematics
- **Prerequisites:** DSC-2 Real Analysis, DSC-5 Calculus and DSC-8 Riemann Integration
- **Covers:** Functions of several variables, partial derivatives, gradient, extrema, Lagrange multipliers, double and triple integrals, change of variables, line and surface integrals, Green's, Stokes' and Gauss' theorems
- **Practice here:** [Calculus 3](/learn/calculus-3): partial derivatives, multiple integrals and vector calculus

## Key takeaways

- DSC-11 Multivariate Calculus is a 4 credit Semester IV core paper in the DU B.Sc. (Hons.) Mathematics programme under UGCF 2022.
- Its prerequisites are Real Analysis, Calculus (DSC-5) and Riemann Integration (DSC-8).
- The three units are differential calculus of several variables, double and triple integrals, and the theorems of Green, Stokes and Gauss.
- The essential reading is Strauss, Bradley and Smith, Calculus (3rd edition).

**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](/courses/university-of-delhi-dsc-5) and [DSC-8 Riemann Integration](/courses/university-of-delhi-dsc-8). This guide follows the official syllabus from the DU Department of Mathematics.

## Course at a glance

| Item | Detail |
| --- | --- |
| Semester | IV |
| Credits | 4 (3 lecture + 1 tutorial) |
| Eligibility | Class XII pass with Mathematics |
| Prerequisites | DSC-2 Real Analysis, DSC-5 Calculus, DSC-8 Riemann Integration |
| Essential reading | Strauss, Bradley and Smith, Calculus (3rd edition) |

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

| DSC-11 topic | Practice | Read |
| --- | --- | --- |
| Functions of several variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives](/blog/partial-derivatives) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Clairaut's theorem](/theorems/clairauts-theorem) |
| Tangent planes and total differential | [Tangent planes](/learn/calculus-3/partial-derivatives/tangent-planes) | [Linear approximation](/blog/linear-approximation) |
| Chain rules | [Multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Partial derivatives practice](/practice-problems/partial-derivatives) |
| Directional derivatives and gradient | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Extrema of two variables | [Second derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Optimization practice](/practice-problems/optimization) |
| Lagrange multipliers | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Double integrals over general regions | [General regions](/learn/calculus-3/multiple-integrals/general-regions) | [Double integrals](/blog/double-integrals) |
| Polar double integrals | [Polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals) | [Double integrals practice](/practice-problems/double-integrals) |
| Triple integrals and volume | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) | [Multiple integrals summary](/learn/calculus-3/multiple-integrals/multiple-integrals-summary) |
| Cylindrical and spherical coordinates | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Points and distance](/learn/calculus-3/vectors-and-space/points-and-distance) |
| Change of variables | [Change of variables](/learn/calculus-3/multiple-integrals/change-of-variables) | [Switching the order](/learn/calculus-3/multiple-integrals/switching-the-order) |
| Line integrals, mass and work | [Line integrals of functions](/learn/calculus-3/vector-calculus/line-integrals-of-functions) | [Line integrals of vector fields](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) |
| Conservative fields | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) | [Vector fields](/learn/calculus-3/vector-calculus/vector-fields) |
| Green's theorem | [Green's theorem](/learn/calculus-3/vector-calculus/greens-theorem) | [Green's theorem](/theorems/greens-theorem) |
| Surface integrals, Stokes and Gauss | [Stokes and divergence theorems](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Stokes' theorem](/theorems/stokes-theorem), [divergence theorem](/theorems/divergence-theorem) |

## 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

$$\int_0^{2\pi}\int_0^{\pi}\int_0^1 \rho^2\sin\phi\,d\rho\,d\phi\,d\theta = 2\pi \cdot 2 \cdot \frac{1}{3} = \frac{4\pi}{3}$$

**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](/practice) and keep the [formula sheet](/formulas) close for coordinate changes.

## Frequently asked questions

### What is DSC-11 at Delhi University?

DSC-11 Multivariate Calculus is a Discipline Specific Core course in Semester IV of the University of Delhi B.Sc. (Hons.) Mathematics programme under UGCF 2022. It has 3 lectures and 1 tutorial a week for 4 credits and extends single variable calculus to functions of two and three variables.

### What are the units of DU Multivariate Calculus?

Unit I (18 hours) is calculus of functions of several variables, Unit II (15 hours) is double and triple integrals, and Unit III (12 hours) is Green's, Stokes' and the Gauss divergence theorem with vector fields, line integrals and surface integrals.

### Which textbook is used for DSC-11?

The syllabus lists Strauss, Bradley and Smith, Calculus (3rd edition, Pearson) as the essential reading and Marsden, Tromba and Weinstein, Basic Multivariable Calculus, as suggested reading.

### How should I prepare for DSC-11?

Revise single variable differentiation and integration, then practise partial derivatives until they are routine. For Unit II, always sketch the region. For Unit III, learn when a field is conservative, because it turns many line integrals into one subtraction.