# Southampton MATH1065 Calculus II: Syllabus, Topics and Practice

> A study guide for MATH1065 Calculus II at Southampton: inhomogeneous ODEs, partial derivatives, Lagrange multipliers and double integrals, with free practice.

Source: https://practicecalculus.com/courses/university-of-southampton-math1065
Updated: 2026-11-08

## At a glance

- **Module code:** MATH1065
- **Credits:** 15 CATS points, Level 4 (Year 1)
- **Semester:** Semester 2
- **Prerequisites:** Not stated on the module page; it follows MATH1064 Calculus I
- **Covers:** Second order inhomogeneous ODEs, variation of parameters, boundary value problems, functions of two variables, partial derivatives, gradient, tangent planes, chain rule and Jacobians, critical points, Lagrange multipliers, multivariable Taylor series, double integrals
- **Assessment:** 20% coursework, 80% exam
- **Previous module:** MATH1064 Calculus I
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH1065 Calculus II is a 15-credit Level 4 module taught in Semester 2 at the University of Southampton.
- It extends calculus to several variables, a key step towards modelling complex systems with differential equations.
- It opens with second order inhomogeneous ODEs, then covers partial derivatives, critical points, Lagrange multipliers and double integrals.
- It is assessed by 20% coursework and an 80% exam.

**MATH1065 Calculus II** is the Semester 2 calculus module for Year 1 mathematics students at the University of Southampton. It follows [MATH1064 Calculus I](/courses/university-of-southampton-math1064) and extends calculus to functions of several variables. Southampton calls this a key step towards formulating differential equations for complex systems, a core part of modelling in the mathematical sciences. This guide walks through the official syllabus and links each topic to free practice.

## What Southampton MATH1065 covers

The 2026-27 module page lists this syllabus:

- Solution of second order inhomogeneous ODEs, variation of parameters and boundary value problems.
- Sketching functions of two variables and limits of multivariable functions.
- Partial derivatives, the gradient and the directional derivative.
- Tangent planes, maximum and minimum points.
- Change of variables and the chain rule, the Jacobian, and polar coordinates.
- Second and higher partial derivatives, including the chain rule for second derivatives.
- Critical points and their classification, and Lagrange multipliers.
- Multivariable Taylor series.
- Double and repeated integrals, including changing the order in a repeated integral.

The learning outcomes say you should be able to set up and evaluate double integrals over simple regions, classify critical points of two-variable functions, evaluate and interpret derivatives of functions of two or more variables, solve second order inhomogeneous ODEs, and switch coordinate systems for integrals when it helps.

Most of this matches the partial derivatives and double integrals parts of a **Calculus 3** course. Vector calculus (line integrals, Green's and Stokes' theorems) is not on the MATH1065 syllabus.

## MATH1065 topic map

| MATH1065 topic | Practice | Read |
| --- | --- | --- |
| Functions of two variables, sketching and limits | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives guide](/blog/partial-derivatives) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivative problems](/practice-problems/partial-derivatives) |
| Gradient and directional derivative | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient guide](/blog/gradient-and-directional-derivatives) |
| Tangent planes | [Tangent planes](/learn/calculus-3/partial-derivatives/tangent-planes) | [Linear approximation](/blog/linear-approximation) |
| Chain rule and change of variables | [Multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Chain rule](/rules/chain-rule) |
| Second and higher partial derivatives | [Second partial derivatives](/learn/calculus-3/partial-derivatives/second-partial-derivatives) | [Clairaut's theorem](/theorems/clairauts-theorem) |
| Critical points and their classification | [Critical points and the second derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Partial derivatives guide](/blog/partial-derivatives) |
| Lagrange multipliers | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers guide](/blog/lagrange-multipliers) |
| Multivariable Taylor series | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) | [Taylor's theorem](/theorems/taylors-theorem) |
| Double and repeated integrals | [Iterated integrals](/learn/calculus-3/multiple-integrals/iterated-integrals) | [Double integrals](/blog/double-integrals) |
| Changing the order of integration | [Switching the order](/learn/calculus-3/multiple-integrals/switching-the-order) | [Double integral problems](/practice-problems/double-integrals) |
| Polar coordinates and the Jacobian | [Polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals) | [Change of variables](/learn/calculus-3/multiple-integrals/change-of-variables) |

The ODE part of the module (variation of parameters and boundary value problems) does not have a matching lesson here yet, so use your problem classes and the textbooks for that block.

## Example problems at the MATH1065 level

**1. Variation of parameters, the setup.** For $y'' + y = \sec x$, the complementary function is $y_c = A\cos x + B\sin x$. Variation of parameters looks for $y_p = u_1(x)\cos x + u_2(x)\sin x$, which gives $y_p = x\sin x + \cos x \ln|\cos x|$.

**2. Classifying a critical point.** For $f(x, y) = x^3 - 3x + y^2$, the critical points are $(1, 0)$ and $(-1, 0)$. At $(1, 0)$, $f_{xx} f_{yy} - f_{xy}^2 = 6 \cdot 2 > 0$ with $f_{xx} > 0$, so it is a local minimum. At $(-1, 0)$ the determinant is negative, so it is a saddle point.

**3. A Lagrange multiplier problem.** Maximise $f = xy$ subject to $x + y = 10$. Solving $\nabla f = \lambda \nabla g$ gives $y = x = 5$ and a maximum value of $25$.

**4. Changing the order of integration.**

$$\int_0^1 \int_x^1 e^{y^2}\,dy\,dx = \int_0^1 \int_0^y e^{y^2}\,dx\,dy = \int_0^1 y e^{y^2}\,dy = \frac{e - 1}{2}$$

## How to do well in MATH1065

- **Prepare for a heavy exam.** The exam is worth 80%, so build a revision plan that covers every syllabus line, not just the topics from coursework.
- **Sketch regions before integrating.** Most errors in repeated integrals come from wrong limits, especially when you change the order.
- **Revisit your ODE methods from MATH1064.** Variation of parameters builds directly on complementary functions and particular integrals.
- **Practise the second derivative test** until you can classify a critical point in under a minute.
- **Check your algebra.** Our list of [common calculus mistakes](/blog/common-calculus-mistakes) covers the slips that cost the most marks.

> **Tip:** the multivariable chain rule is the idea behind the Jacobian, polar coordinates and change of variables. Get it right in the [chain rule lesson](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) and several syllabus lines fall into place at once.

## Frequently asked questions

### What is MATH1065 at Southampton?

MATH1065 Calculus II is the Semester 2 calculus module for first-year mathematics students. Southampton says it extends the ideas of calculus to multiple variables, which is a key step towards formulating differential equations for complex systems.

### Is Southampton Calculus II like a US Calculus 2 course?

Not really. A US Calculus 2 course covers integration techniques and series in one variable, and Southampton teaches much of that in MATH1064. MATH1065 is mostly multivariable calculus, closer to a Calculus 3 course, with extra work on ODEs.

### How is MATH1065 assessed?

The 2026-27 module page lists 20% coursework and an 80% exam. Teaching is 48 hours of lectures, problem classes and workshops, plus 102 hours of independent study.

### Which textbooks does MATH1065 use?

The module page lists Calculus: A Complete Course by R. A. Adams and Advanced Calculus by R. Wrede and M. R. Spiegel (Schaum's series).

### How can I practise for MATH1065?

Practise partial derivatives, the second derivative test, Lagrange multipliers and double integrals one skill at a time. The free Calculus 3 path on PracticeCalculus covers each with instant feedback.