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 [https://en.wikipedia.org/wiki/National_qualifications_frameworks_in_the_United_Kingdom] 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 OUDCE '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.