Oxford Introductory Calculus (Prelims M3): Syllabus and Practice

Introductory Calculus is the first applied calculus course in the Oxford Mathematics Prelims, the first-year course for the BA/MMath. It runs for 16 lectures in Michaelmas Term and is examined in paper Mathematics III. This guide sets out what the official 2025-26 Prelims synopses say about the course and links every topic to free practice, so you can learn each idea by solving problems with instant feedback.

What Oxford Introductory Calculus covers

The synopses describe the course as a gentle introduction to applied mathematics in your first term, with an "A level" feel, to help you move from school to university. The emphasis is on building skill and familiarity through straightforward examples. By the end, you should be able to solve a range of ordinary differential equations (ODEs) and evaluate partial derivatives and use them in applications.

The syllabus has six parts:

1. Linear ODEs. Integrating factors for first order linear equations, a second solution when one is known, constant coefficient equations, and general solutions as particular integral plus complementary function.
2. Partial derivatives. Second order derivatives, equality of mixed partials, the chain rule, change of variables including plane polar coordinates, and simple PDEs such as f_{xy} = 0.
3. Curves. Parametric curves, tangents, arc length and line integrals.
4. Jacobians and double integrals. Area calculations and the classic integral of e^{-(x^2+y^2)} over the plane.
5. Surfaces and the gradient. Level sets, normals, directional derivatives, and \int_A^B \nabla\phi \cdot d\mathbf{r} = \phi(B) - \phi(A).
6. Critical points. Taylor's theorem in two variables (statement only), classifying critical points, and an informal, geometric look at Lagrange multipliers.

Compared with a North American sequence, this is mostly Calculus 3 material (partial derivatives, line and double integrals) together with the first-order ODEs found at the end of Calculus 2.

Introductory Calculus topic map

PracticeCalculus does not yet have lessons on second order constant coefficient ODEs, so use your problem sheets for that part and the lessons above for everything else.

Example problems at the Introductory Calculus level

1. An integrating factor. Solve y' + 2y = e^{x}. Multiply by e^{2x}:

2. A mixed partial. For f(x, y) = x^2 y^3, we get f_x = 2xy^3 and

3. The Gaussian integral. Switching to polar coordinates, where dA = r\,dr\,d\theta:

How to do well in Introductory Calculus

• Keep your A level calculus sharp. Integrating factors and double integrals lean on integration by parts and u-substitution all the time.
• Do a little every day. Tutorials come round fast in Michaelmas, and short daily sessions on partial derivatives build speed better than one long session before a problem sheet is due.
• Draw the picture. For line integrals, Jacobians and Lagrange multipliers, a quick sketch of the curve, region or level set usually shows the method.
• Practise under exam conditions before Prelims with mixed practice exams.

Tip: the course hands straight over to M5: Multivariable Calculus in Hilary Term. See the Oxford Multivariable Calculus guide for what comes next.