# UNSW MATH2011 Several Variable Calculus: Study Guide, Practice

> A study guide for UNSW MATH2011 Several Variable Calculus: gradients, Lagrange multipliers, multiple integrals, line and surface integrals and Stokes.

Source: https://practicecalculus.com/courses/unsw-math2011
Updated: 2026-11-03

## At a glance

- **Course code:** MATH2011
- **Units of credit:** 6
- **Offered:** Term 1
- **Prerequisites:** MATH1231 or MATH1241 or MATH1251 or DPST1014
- **Covers:** Functions of several variables, limits, differentiability, gradients, extrema, Taylor series, Lagrange multipliers, Jacobians, double and triple integrals, line and surface integrals, divergence and curl, Green and Stokes
- **Higher version:** MATH2111 (Higher Several Variable Calculus)
- **Previous course:** MATH1231 (Mathematics 1B)
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH2011 Several Variable Calculus is a 6 unit of credit second-year course at UNSW Sydney, offered in Term 1.
- It covers differential, integral and vector calculus for functions of more than one variable, including Green's and Stokes' theorems.
- You need MATH1231, MATH1241, MATH1251 or DPST1014 to enrol.
- MATH2111 is the higher version, with the same topics treated with more theory; the two courses exclude each other.

**MATH2011 Several Variable Calculus** is UNSW Sydney's second-year course in multivariable calculus, taken after Mathematics 1B by students in mathematics, physics, science and other quantitative programs. It is worth 6 units of credit and runs in Term 1. This guide maps every MATH2011 topic to free practice, so you can build each skill by solving problems with instant feedback.

## What MATH2011 covers

The UNSW Handbook describes MATH2011 as an introduction to differential, integral and vector calculus for functions of more than one variable. The listed topics fall into three blocks:

1. **Differential calculus:** functions of several variables, limits and continuity, differentiability, gradients, surfaces, maxima and minima, Taylor series, Lagrange multipliers, chain rules, the inverse function theorem and Jacobian derivatives.
2. **Integral calculus:** double and triple integrals, iterated integrals, Riemann sums, cylindrical and spherical coordinates, change of variables and centre of mass.
3. **Vector calculus:** line integrals, parametrised surfaces, surface integrals, del, divergence and curl, Stokes' theorem, Green's theorem in the plane, and applications to fluid dynamics and electrodynamics.

That is a complete standard **Calculus 3** course, including the vector calculus that some universities teach separately.

## MATH2011 topic map

| MATH2011 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) |
| Differentiability and tangent planes | [Tangent planes and linear approximation](/learn/calculus-3/partial-derivatives/tangent-planes) | [Linear approximation](/blog/linear-approximation) |
| Gradients | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Maxima and minima | [Critical points and the second-derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Clairaut's theorem](/theorems/clairauts-theorem) |
| Taylor series in several variables | [Second partial derivatives](/learn/calculus-3/partial-derivatives/second-partial-derivatives) | [Taylor's theorem](/theorems/taylors-theorem) |
| Lagrange multipliers | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Chain rules | [Multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Chain rule](/rules/chain-rule) |
| Double and iterated integrals | [Double integrals over rectangles](/learn/calculus-3/multiple-integrals/iterated-integrals) | [Double integrals](/blog/double-integrals) |
| Triple integrals, cylindrical and spherical | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) |
| Change of variables and Jacobians | [Change of variables and the Jacobian](/learn/calculus-3/multiple-integrals/change-of-variables) | [Double integrals](/blog/double-integrals) |
| Centre of mass | [Area, volume, mass and average value](/learn/calculus-3/multiple-integrals/double-integral-applications) | [Center of mass](/learn/calculus-2/improper-integrals-and-applications/center-of-mass) |
| Line integrals | [Line integrals of vector fields](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) | [Line integrals of functions](/learn/calculus-3/vector-calculus/line-integrals-of-functions) |
| Divergence and curl | [Curl and divergence](/learn/calculus-3/vector-calculus/curl-and-divergence) | [Divergence Theorem](/theorems/divergence-theorem) |
| Surface integrals | [Surface integrals and flux](/learn/calculus-3/vector-calculus/surface-integrals-and-flux) | [Vectors, dot and cross product](/blog/vectors-dot-and-cross-product) |
| Green's and Stokes' theorems | [Green's theorem](/learn/calculus-3/vector-calculus/greens-theorem) | [Green's theorem](/theorems/greens-theorem), [Stokes' theorem](/theorems/stokes-theorem) |

## MATH2011 vs MATH2111

| | MATH2011 Several Variable Calculus | MATH2111 Higher Several Variable Calculus |
| --- | --- | --- |
| Units of credit | 6 | 6 |
| Term | Term 1 | Term 1 |
| Prerequisite | MATH1231, MATH1241, MATH1251 or DPST1014 | One of those with a mark of at least 70 |
| Style | Methods and applications | Core higher course, more theory |

The two courses list nearly the same topics and exclude each other. Engineering students often take engineering mathematics courses instead, such as MATH2018 or MATH2019, which cover partial differentiation, double integrals and an introduction to vector field theory alongside other topics. If Mathematics 1B feels distant, review the [MATH1231 guide](/courses/unsw-math1231) and the [MATH1131 guide](/courses/unsw-math1131).

## Example problems at the MATH2011 level

**1. Classifying a critical point.** For $f(x, y) = x^2 + xy + y^2$, $f_x = 2x + y$ and $f_y = x + 2y$ are both zero only at $(0, 0)$. With $f_{xx} = 2$, $f_{yy} = 2$ and $f_{xy} = 1$, the discriminant is $D = 2 \cdot 2 - 1^2 = 3 > 0$ and $f_{xx} > 0$, so $(0, 0)$ is a local minimum.

**2. Divergence and curl.** For $\mathbf{F} = \langle xy, yz, zx \rangle$:

$$\nabla \cdot \mathbf{F} = y + z + x, \qquad \nabla \times \mathbf{F} = \langle -y, -z, -x \rangle$$

**3. A spherical integral.** The volume of a ball of radius $a$ is

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

## How to do well in MATH2011

- **Master the gradient early.** It drives directional derivatives, tangent planes and Lagrange multipliers. Our [gradient guide](/blog/gradient-and-directional-derivatives) ties them together.
- **Sketch regions and surfaces** before setting up any integral, and practice reordering with the [switching the order](/learn/calculus-3/multiple-integrals/switching-the-order) lesson.
- **Learn Green, Stokes and the Divergence Theorem as one idea:** a boundary integral equals an interior integral of a derivative. Compare the [Green's theorem](/theorems/greens-theorem) and [Stokes' theorem](/theorems/stokes-theorem) pages side by side.
- **Keep the Jacobian factors** for polar, cylindrical and spherical coordinates on your [formula sheet](/formulas), and rehearse with timed [practice](/practice) sets.

> **Tip:** if a line integral looks messy, compute the curl first. A zero curl on a simply connected region means the field is conservative, and the [potential function](/learn/calculus-3/vector-calculus/conservative-fields) does the rest.

## Frequently asked questions

### What is MATH2011 at UNSW?

MATH2011 Several Variable Calculus is UNSW Sydney's second-year course in multivariable calculus. It covers differentiation of functions of several variables, Lagrange multipliers and Jacobians, double and triple integrals, and vector calculus up to Green's and Stokes' theorems.

### What is the difference between MATH2011 and MATH2111?

MATH2111 Higher Several Variable Calculus is the higher version. The Handbook describes it as extending the theory of one-variable calculus to several variables, with a very similar topic list, and it requires a mark of at least 70 in MATH1231 or an equivalent. The two courses exclude each other.

### What are the prerequisites for MATH2011?

The Handbook lists MATH1231, MATH1241, MATH1251 or DPST1014.

### Is MATH2011 hard?

It brings together a lot of material in one term, from partial derivatives to surface integrals. Students who already have the partial derivatives and chain rules from MATH1231 under control, and who sketch every region, find it much more approachable.

### How should I prepare for MATH2011?

Revise partial derivatives and the multivariable chain rule from MATH1231, plus vectors and polar coordinates. The Calculus 3 path on PracticeCalculus covers these first.