Cambridge Part IA Vector Calculus: Syllabus, Topics and Practice

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

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

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,

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

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

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.