Southampton MATH1065 Calculus II: Syllabus, Topics and Practice

MATH1065 Calculus II is the Semester 2 calculus module for Year 1 mathematics students at the University of Southampton. It follows MATH1064 Calculus I 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

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.

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 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 and several syllabus lines fall into place at once.