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 ‘Calculus: Introduction’ 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.

Course Details

This course begins on the 15 Sep 2026 which is when course materials are made available to students. Students should study these materials in advance of the first live meeting which will be held on 22 Sep 2026, 7:30-8:30pm (UK time).

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

Course aims

• To develop a deeper knowledge of Calculus through the study of Multi-Variable Functions and Partial Differential Equations. Follows OUDCE’s ‘Introduction to Calculus’ course.
• To introduce the concept of mathematical modelling through the vehicle of Calculus.
• To extend student’s knowledge beyond the basics of computation, to an understanding of theory and derivation of formulae.
• To develop the high-level analytical skills required of a Mathematician.

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.

Level and demands

Knowledge of differentiation is required, including the limit definition of a derivative; the rules of differentiation such as the chain rule, product rule and quotient rule; as well as familiarity with vectors, their properties and manipulation. Some basic knowledge of Physics would be helpful, but is not essential.

A typical student on this course will have studied Calculus at high school (at some point in the past) and may have taken an Introductory Calculus course at College or University. This course is designed to follow on from OUDCE’s ‘Introduction to Calculus’ course.

This course is offered at FHEQ Level 4 (i.e. first year undergraduate level), and you will be expected to engage in independent study in preparation for your assignments and for the weekly webinar. This may take the form, for instance, of reading and analysing set texts, responding to questions or tasks, or preparing work to present in class. Our 10-week Short Online Courses come with an expected total commitment of 100 study hours, including those spent in live webinars.

Teaching methods

This course takes place over 10 weeks, with a weekly learning schedule and weekly live webinar held on Microsoft Teams. Shortly before a course commences, students are provided with access to an online virtual learning environment, which houses the course content, including video lectures, complemented by readings or other study materials. Any standard web browser can be used to access these materials, but we recommend Google Chrome. Working through these materials over the course of the week will prepare students for a weekly 1-hour live webinar you will share with your expert tutor and fellow students. All courses are structured to amount to 100 study hours, so that on average, you should set aside 10 hours a week for study. Although the course finishes after 10 weeks, all learning materials remain available to all students for 12 months after the course has finished.

All courses are led by an expert tutor. Tutors guide students through the course materials as part of the live interactions during the weekly webinars. Tutors will also provide individualised feedback on your assignments. All online courses are taught in small student cohorts so that you and your peers will form a mutually supportive and vibrant learning community for the duration of the course. You will learn from your fellow students as well as from your tutor, and they will learn from you.

Assessment methods

You will be set independent formative and summative work for this course. Formative work will be submitted for informal assessment and feedback from your tutor, but has no impact on your final grade. The summative work will be formally assessed as pass or fail.

Certification

All students who pass their final assignment 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.