# Cambridge Part IA Vector Calculus: Syllabus, Topics and Practice

> A study guide for Cambridge Part IA Vector Calculus: line, surface and volume integrals, div, grad and curl, Green, Stokes and the Divergence Theorem.

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

## At a glance

- **Course code:** Part IA Vector Calculus, Mathematical Tripos (Cambridge does not use numeric module codes)
- **Lectures:** 24 lectures
- **Term:** Lent Term, Year 1
- **Examined in:** Paper 3, with Groups
- **Prerequisites:** None listed in the schedules
- **Note:** Omitted from Option (b), Mathematics with Physics
- **Covers:** Curves and arc length, line, surface and volume integrals, gradient, divergence and curl, Green's, Stokes's and the Divergence Theorem, Laplace's equation, Cartesian tensors
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- Vector Calculus is a 24-lecture Lent Term course in Part IA of the Cambridge Mathematical Tripos, examined in Paper 3.
- It covers curves, line, surface and volume integrals, and the operators grad, div and curl.
- Its centre is the integration theorems: the Divergence Theorem, Green's theorem and Stokes's theorem, with applications to fluids and electromagnetism.
- It ends with Laplace's equation and Cartesian tensors, and is not taken by Mathematics with Physics students.

**Vector Calculus** is the multivariable integration course in Part IA, the first year of the Cambridge Mathematical Tripos. It runs for 24 lectures in Lent Term and is examined in Paper 3, alongside Groups. It takes you from curves in space to the big integration theorems and their uses in physics. This guide follows the official 2026-27 Faculty of Mathematics schedules and links every topic to free practice.

## What Cambridge Vector Calculus covers

The schedules list six blocks, with approximate lecture counts:

1. **Curves in $\mathbb{R}^3$ [1].** Parameterised curves, arc length, tangents and normals, curvature and torsion.
2. **Integration in $\mathbb{R}^2$ and $\mathbb{R}^3$ [4].** Line integrals, and surface and volume integrals in Cartesian, cylindrical and spherical coordinates, including change of variables.
3. **Vector operators [5].** Directional derivatives, the gradient as a normal to level surfaces, divergence, curl and $\nabla^2$, solenoidal, irrotational and conservative fields, scalar potentials and vector identities.
4. **Integration theorems [5].** The Divergence Theorem, Green's theorem, Stokes's theorem and Green's second theorem, with applications to fluid dynamics and to electromagnetism, including Maxwell's equations.
5. **Laplace's equation [4].** Uniqueness and the maximum principle, and solving Poisson's equation by Gauss's method and as an integral.
6. **Cartesian tensors [5].** Transformation laws, isotropic tensors, symmetric and antisymmetric tensors, the quotient theorem, and examples such as inertia and conductivity.

The schedules list no formal prerequisites. In North American terms the first four blocks are the second half of **Calculus 3**. Laplace's equation and Cartesian tensors go beyond a standard Calculus 3 course.

## Vector Calculus topic map

| Schedule topic | Practice | Read |
| --- | --- | --- |
| Parameterised curves and tangents | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | [Vectors, dot and cross product](/blog/vectors-dot-and-cross-product) |
| Arc length | [Parametric arc length](/learn/calculus-2/power-series-parametric-polar/parametric-arc-length) | [Parametric and polar problems](/practice-problems/parametric-and-polar) |
| 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) |
| Volume integrals | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) | [Double integrals](/blog/double-integrals) |
| Cylindrical and spherical coordinates | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Double integral problems](/practice-problems/double-integrals) |
| Change of variables | [Change of variables](/learn/calculus-3/multiple-integrals/change-of-variables) | [Polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals) |
| Surface integrals | [Surface integrals and flux](/learn/calculus-3/vector-calculus/surface-integrals-and-flux) | [Vector fields](/learn/calculus-3/vector-calculus/vector-fields) |
| Gradient and directional derivatives | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient guide](/blog/gradient-and-directional-derivatives) |
| Divergence and curl | [Curl and divergence](/learn/calculus-3/vector-calculus/curl-and-divergence) | [Partial derivatives guide](/blog/partial-derivatives) |
| Conservative fields and potentials | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) | [Clairaut's theorem](/theorems/clairauts-theorem) |
| Green's theorem | [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) |

PracticeCalculus does not cover curvature and torsion, Laplace's equation or Cartesian tensors, so use your lecture notes and example sheets for those blocks.

## Example problems at the Part IA level

**1. A line integral.** For $\mathbf{F} = (y, x)$ along $\mathbf{r}(t) = (t, t^2)$ with $0 \le t \le 1$, we get $\mathbf{F}\cdot\mathbf{r}'(t) = t^2 + 2t^2 = 3t^2$, so

$$\int_C \mathbf{F}\cdot d\mathbf{r} = \int_0^1 3t^2\,dt = 1$$

Since $\mathbf{F} = \nabla(xy)$, this also equals $xy$ at $(1, 1)$ minus $xy$ at $(0, 0)$.

**2. A vector identity.** For any smooth scalar field $\phi$,

$$\nabla \times (\nabla \phi) = \mathbf{0}$$

because each component is a difference of equal mixed partial derivatives.

**3. Green's theorem.** For the unit disc $D$ with boundary traversed anticlockwise,

$$\oint_{\partial D} (-y\,dx + x\,dy) = \iint_D 2\,dA = 2\pi$$

## How to do well in Vector Calculus

- **Sketch every region and surface.** Choosing coordinates and limits is most of the work in surface and volume integrals.
- **Learn suffix notation early.** The schedules use it for vector identities and tensors, and it turns long identity proofs into a few lines.
- **Check with both sides of a theorem.** If you compute a flux directly, check it with the Divergence Theorem. The two answers should match.
- **Practise short Section I questions** on div, curl and line integrals until they take minutes, not tens of minutes.

> **Tip:** Differential Equations in Michaelmas introduces the gradient and directional derivatives, which this course builds on. See the [Cambridge Differential Equations guide](/courses/university-of-cambridge-differential-equations).

## Frequently asked questions

### What is Part IA Vector Calculus at Cambridge?

It is the first-year Mathematical Tripos course on calculus in two and three dimensions. The 2026-27 schedules list 24 lectures in Lent Term on curves, multiple and surface integrals, vector operators, the integration theorems, Laplace's equation and Cartesian tensors.

### Do all Part IA students take Vector Calculus?

No. The schedules note that it is omitted from Option (b), Mathematics with Physics. Students on Option (a), Pure and Applied Mathematics, take it and are examined on it in Paper 3.

### Are the integration theorems proved?

The schedules ask for statements, informal proofs and examples of the Divergence Theorem, Green's theorem, Stokes's theorem and Green's second theorem, with applications to fluid dynamics and Maxwell's equations.

### How can I prepare for Vector Calculus?

Be fluent with partial derivatives and multiple integrals in Cartesian, cylindrical and spherical coordinates, then practise div, curl, flux and the integral theorems. The Calculus 3 path follows a similar order.