Learn Calculus at the University of Oxford with this 10-week online course led by Tom. Full information below.

The tools of calculus are essential to our understanding of the universe, and as such they form the basis for the majority of mathematical models, from the spread of disease to the physics of glaciers.

The course begins with the familiar example of Ordinary Differential Equations (ODEs) and revisits some of the most-common methods of solution such as Integrating Factors, Homogeneous Functions, and Separation of Variables. We then use apply the tools of one-variable calculus to solve some problems related to optimisation.

Next, we introduce the concept of multi-variable functions through Partial Differentiation, Taylor’s Theorem, and solving simple PDEs. We will also briefly touch on Vector Calculus in the form of the Gradient Vector and the Divergence.

The latter part of the course focuses on the specific example of the Heat Equation – one of the most fundamental PDEs and the gateway to the method of Fourier Series. 

We will end with a closer look at some applications of our newly discovered techniques to real-world problems, such as disease modelling and ice flow in a glacier, and (time-permitting) a short introduction to Integration and Jacobians.

This is an ‘intermediate’ FHEQ level 4 course and therefore in order to get the most out of the teaching you should have some familiarity with Calculus as a pre-requisite. In particular, a knowledge of differentiation is a must. Taking the Oxford Lifelong Learning ‘Beginning Calculus’ course would be ample preparation.

The overall structure of the course follows the Undergraduate Mathematics Syllabus at the University of Oxford. The topics covered each week are listed below.

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This course combines online study with a weekly 1-hour live webinar (Tuesdays 7:30 – 8:30pm) led by your tutor. Find out more about how our short online courses are taught.

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Programme details

Course Begins: 23 September 2025

Week 1: Differential Equations
Week 2: Solving ODEs
Week 3: Optimisation
Week 4: Partial Differentiation
Week 5: Taylor’s Theorem and Critical Points
Week 6: Partial Differential Equations
Week 7: Heat Equation
Week 8: Fourier Series
Week 9: Applications
Week 10: Integrals  

All students who pass their final assignment, whether registered for credit or not, will be eligible for a digital Certificate of Completion. Upon successful completion, you will receive a link to download a University of Oxford digital certificate. Information on how to access this digital certificate will be emailed to you after the end of the course. The certificate will show your name, the course title and the dates of the course you attended. You will be able to download your certificate or share it on social media if you choose to do so. 

Please note that assignments are not graded but are marked either pass or fail.

Course aims

Develop a deeper knowledge of Calculus through the study of Multi-Variable Functions and Partial Differential Equations. Follows the ‘Introduction to Calculus’ course.

Course Objectives

• Introduce the concept of mathematical modelling through the vehicle of Calculus;
• Extend student’s knowledge beyond the basics of computation, to an understanding of theory and derivation of formulae;
• Develop the high-level analytical skills required of a Mathematician.

Teaching methods

The course will consist of the following:
– A weekly lecture video (averaging 50-60 minutes over the 10 weeks) to cover the core concepts of each topic
– Guided reading of lecture notes, textbooks, sample exercises
– A weekly problem set
– A 1-hour weekly group tutorial to cover the solutions to the problem set and answer any questions about the content

Learning outcomes

By the end of the course students will be expected to:
– Solve simple first and second order ODEs using the techniques of Integrating Factors, Homogenous Functions, and Separation of Variables;
– Apply the tools of Calculus to develop Mathematical Models for a variety of real-world situations, including optimisation problems;
– Demonstrate an understanding of multi-variable functions through Partial Differentiation and Taylor Series, as well as employing the techniques of Separable Solutions and Fourier Series to solve PDEs.

Assessment methods

Weekly problem sets which will be used to determine the content of the tutorials. A ‘mock’ exam in week 5 as a formative assessment and practice for the final exam at the end of the course. The final exam will be untimed, open-book and will cover all topics in the course. It will determine the final grade.

Coursework is an integral part of all weekly classes and everyone enrolled will be expected to do coursework in order to benefit fully from the course. Only those who have registered for credit will be awarded CATS points for completing work the required standard.