# Oxford Multivariable Calculus (Prelims M5): Syllabus and Practice

> A study guide for Oxford Prelims Multivariable Calculus: multiple integrals, div, grad and curl, the Divergence Theorem and Stokes, with free practice.

Source: https://practicecalculus.com/courses/university-of-oxford-multivariable-calculus
Updated: 2026-11-08

## At a glance

- **Course code:** M5: Multivariable Calculus (examined in Prelims paper Mathematics V; Oxford does not use numeric module codes)
- **Lectures:** About 16 (the synopsis allocates 16 lectures across its topics)
- **Term:** Hilary Term, Year 1
- **Prerequisites:** Not listed formally; it opens with a recap of line and surface integrals from Introductory Calculus
- **Covers:** Double and volume integrals, Jacobians, flux and work integrals, divergence and curl, the Divergence Theorem, Stokes's theorem, Gauss' flux theorem
- **Paired course:** M5: Fourier Series and PDEs
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- Multivariable Calculus is a first-year Oxford Prelims course examined in paper Mathematics V, alongside Fourier Series and PDEs.
- It covers multiple integrals, div, grad and curl, and the integral theorems of vector calculus.
- The synopses describe the Divergence Theorem and Stokes's theorem as higher dimensional versions of the Fundamental Theorem of Calculus.
- It builds on the line integrals, gradient and Jacobians from Introductory Calculus.

**Multivariable Calculus** is the vector calculus course in the Oxford Mathematics Prelims, the first year of the BA/MMath. It is taught in Hilary Term and examined in paper Mathematics V, which it shares with Fourier Series and PDEs. This guide summarises the official 2025-26 Prelims synopses and links every topic to free practice, so you can build fluency with integrals and integral theorems by solving problems.

## What Oxford Multivariable Calculus covers

The synopses say the course introduces multi-dimensional vector calculus. You learn to evaluate volume, surface and line integrals in three dimensions and see how they are related by the Divergence Theorem and Stokes's theorem, which the synopses describe as higher dimensional versions of the Fundamental Theorem of Calculus. You should also be able to calculate with div, grad and curl, understand what they mean physically, and prove the main identities.

The synopsis gives the approximate number of lectures for each block:

| Block | Lectures |
| --- | --- |
| Double integrals: repeated integration, general domains, change of variables | 2 |
| Volume integrals with cylindrical and spherical Jacobians | 1.5 |
| Surface and line integrals, flux (including solid angle), work and conservative fields | 2 |
| Scalar and vector fields, divergence and curl, identities | 2.5 |
| Divergence Theorem, Green's theorems, Poisson's equation, the heat equation | 4 |
| Stokes's theorem and potentials for conservative forces | 2 |
| Gauss' flux theorem and Poisson's equation | 2 |

Compared with a North American sequence, this is the second half of **Calculus 3**: multiple integrals and vector calculus. It assumes partial derivatives, the gradient and line integrals from Introductory Calculus.

## Multivariable Calculus topic map

| Syllabus topic | Practice | Read |
| --- | --- | --- |
| Repeated integration over a rectangle | [Iterated integrals](/learn/calculus-3/multiple-integrals/iterated-integrals) | [Double integrals](/blog/double-integrals) |
| General domains | [General regions](/learn/calculus-3/multiple-integrals/general-regions) | [Switching the order](/learn/calculus-3/multiple-integrals/switching-the-order) |
| Change of variables in two dimensions | [Change of variables](/learn/calculus-3/multiple-integrals/change-of-variables) | [Polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals) |
| Volume integrals | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) | [Double integral problems](/practice-problems/double-integrals) |
| Cylindrical and spherical Jacobians | [Cylindrical and spherical coordinates](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Change of variables](/learn/calculus-3/multiple-integrals/change-of-variables) |
| Line and work 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) |
| Conservative fields and potentials | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) | [Gradient guide](/blog/gradient-and-directional-derivatives) |
| Surface and flux integrals | [Surface integrals and flux](/learn/calculus-3/vector-calculus/surface-integrals-and-flux) | [Vector fields](/learn/calculus-3/vector-calculus/vector-fields) |
| Divergence and curl | [Curl and divergence](/learn/calculus-3/vector-calculus/curl-and-divergence) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Divergence theorem in the plane | [Green's theorem](/learn/calculus-3/vector-calculus/greens-theorem) | [Green's theorem](/theorems/greens-theorem) |
| Divergence Theorem | [Stokes and divergence theorems](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Divergence Theorem](/theorems/divergence-theorem) |
| Stokes's theorem | [Stokes and divergence theorems](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Stokes' theorem](/theorems/stokes-theorem) |

Poisson's equation, the heat equation and Gauss' flux theorem are applications that PracticeCalculus does not cover directly, so use your lecture notes and problem sheets for those.

## Example problems at the Multivariable Calculus level

**1. A volume integral in spherical polars.** The volume of a ball of radius $a$, with $dV = r^2\sin\theta\,dr\,d\theta\,d\phi$:

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

**2. Divergence and curl.** For $\mathbf{F} = (x^2, xy, z)$:

$$\nabla\cdot\mathbf{F} = 2x + x + 1 = 3x + 1, \qquad \nabla\times\mathbf{F} = (0, 0, y)$$

**3. The Divergence Theorem.** For $\mathbf{F} = \mathbf{r} = (x, y, z)$ we have $\nabla\cdot\mathbf{F} = 3$, so the outward flux through the sphere of radius $a$ is

$$\iint_S \mathbf{F}\cdot d\mathbf{S} = \iiint_V 3\,dV = 4\pi a^3$$

## How to do well in Multivariable Calculus

- **Master the limits of integration.** Most mistakes in multiple integrals come from the set-up, not the integration. Sketch the region every time.
- **Know your Jacobians by heart.** $r$ for plane polars, $r$ for cylindrical and $r^2\sin\theta$ for spherical polars appear in almost every question.
- **Choose the easier side of each theorem.** A flux integral over a closed surface is often much quicker as a volume integral of the divergence, and the reverse can also be true.
- **Practise mixed questions** before Prelims with [practice exams](/practice), and review [common calculus mistakes](/blog/common-calculus-mistakes).

> **Tip:** if div and curl feel abstract, start with [vector fields](/learn/calculus-3/vector-calculus/vector-fields) and picture each field as a flow. Divergence measures how much it spreads out and curl how much it swirls.

## Frequently asked questions

### What is Multivariable Calculus at Oxford?

It is the Prelims course on vector calculus in three dimensions. The 2025-26 synopses say students learn to evaluate volume, surface and line integrals and see how they are related through the Divergence Theorem and Stokes's theorem.

### What do I need to know before Multivariable Calculus?

The synopsis starts with a recap of surface and line integrals, so it assumes Introductory Calculus from Michaelmas: partial derivatives, the gradient, line integrals and Jacobians. Vector algebra from the Geometry course also helps.

### Is the proof of the Divergence Theorem examined?

The synopses say the proof for convex regions is non-examinable. You are expected to use the theorem and its consequences, such as Green's first and second theorems and uniqueness for Poisson's equation.

### How can I practise for Oxford Multivariable Calculus?

Work through multiple integrals in Cartesian, cylindrical and spherical coordinates until setting up limits is routine, then practise div, curl, flux and the integral theorems. The Calculus 3 path follows the same order.