# Oxford Introductory Calculus (Prelims M3): Syllabus and Practice

> A study guide for Oxford Prelims Introductory Calculus: ODEs, partial derivatives, line and double integrals and Lagrange multipliers, with free practice.

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

## At a glance

- **Course code:** M3: Introductory Calculus (examined in Prelims paper Mathematics III; Oxford does not use numeric module codes)
- **Lectures:** 16 lectures
- **Term:** Michaelmas Term, Year 1
- **Prerequisites:** None listed in the synopses; the course is designed to feel like A level and ease the move from school
- **Covers:** Linear ODEs, partial derivatives and the chain rule, curves and line integrals, Jacobians and double integrals, gradient, critical points, Lagrange multipliers
- **Next course:** M5: Multivariable Calculus (Hilary Term)
- **Practice here:** [Calculus 2](/learn/calculus-2) and [Calculus 3](/learn/calculus-3)

## Key takeaways

- Introductory Calculus is a 16-lecture Michaelmas Term course in the first year (Prelims) of the Oxford Mathematics degree.
- The synopses describe it as a gentle introduction to applied mathematics with an "A level feel", to help with the move from school to university.
- It covers linear ODEs, partial derivatives, line integrals, simple double integrals, the gradient and Lagrange multipliers.
- It is examined in Prelims paper Mathematics III, and M5: Multivariable Calculus builds on it later in the year.

**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

| Syllabus topic | Practice | Read |
| --- | --- | --- |
| First order linear and separable ODEs | [Separable equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [Differential equations problems](/practice-problems/differential-equations) |
| Growth models from ODEs | [Growth and decay](/learn/calculus-2/improper-integrals-and-applications/growth-and-decay) | [Exponential functions](/functions/exponential-functions) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivatives guide](/blog/partial-derivatives) |
| Second order and mixed partials | [Second partial derivatives](/learn/calculus-3/partial-derivatives/second-partial-derivatives) | [Clairaut's theorem](/theorems/clairauts-theorem) |
| Chain rule and change of variables | [Multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Chain rule](/rules/chain-rule) |
| Parametric curves and 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 functions](/learn/calculus-3/vector-calculus/line-integrals-of-functions) | [Vectors, dot and cross product](/blog/vectors-dot-and-cross-product) |
| Jacobians and double integrals | [Polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals) | [Double integrals](/blog/double-integrals) |
| Gradient and directional derivatives | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient guide](/blog/gradient-and-directional-derivatives) |
| Gradient fields and potentials | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) | [Line integrals of vector fields](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) |
| Critical points | [Second derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Taylor's theorem](/theorems/taylors-theorem) |
| Lagrange multipliers | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers guide](/blog/lagrange-multipliers) |

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 $e^{2x}$:

$$\frac{d}{dx}\left(e^{2x}y\right) = e^{3x} \quad\Rightarrow\quad y = \tfrac{1}{3}e^{x} + Ce^{-2x}$$

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

$$f_{xy} = 6xy^2 = f_{yx}$$

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

$$\iint_{\mathbb{R}^2} e^{-(x^2+y^2)}\,dA = \int_0^{2\pi}\int_0^{\infty} e^{-r^2} r\,dr\,d\theta = \pi$$

## How to do well in Introductory Calculus

- **Keep your A level calculus sharp.** Integrating factors and double integrals lean on [integration by parts](/blog/integration-by-parts) and [u-substitution](/blog/u-substitution) all the time.
- **Do a little every day.** Tutorials come round fast in Michaelmas, and short daily sessions on [partial derivatives](/learn/calculus-3/partial-derivatives/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](/practice).

> **Tip:** the course hands straight over to M5: Multivariable Calculus in Hilary Term. See the [Oxford Multivariable Calculus guide](/courses/university-of-oxford-multivariable-calculus) for what comes next.

## Frequently asked questions

### What is Introductory Calculus at Oxford?

It is the first applied calculus course in Oxford Prelims, listed under M3. According to the 2025-26 synopses, students learn to solve a range of ordinary differential equations and to evaluate partial derivatives and use them in applications.

### How many lectures is Oxford Introductory Calculus?

The course information page lists 16 lectures in Michaelmas Term. The synopsis splits them roughly as 4 on ODEs, 3.5 on partial derivatives, 1 on curves and line integrals, 2 on Jacobians and double integrals, 2 on surfaces and the gradient, and 3.5 on Taylor's theorem, critical points and Lagrange multipliers.

### Is Introductory Calculus hard?

The synopses say it is meant to be a gentle start with straightforward examples. It moves quickly, though, and partial derivatives and Lagrange multipliers will be new to most students, so regular practice helps.

### What should I revise before starting?

A level differentiation and integration, especially the chain rule, integration by parts and substitution. These skills are used in almost every ODE and multivariable problem in the course.