Over the last four hundred years progress in understanding the physical world (theoretical physics) has gone hand in hand with progress in the mathematical sciences, so much so that the terms applied mathematics and theoretical physics have come to be almost coterminous. Vector calculus is one of the main mathematical tools to study the world around us. Many physical quantities are described by vector or scalar fields. Examples include not only velocities and forces (particularly useful in fluid mechanics), but also particle displacements (useful in solid mechanics), and electric and magnetic fields (electromagnetism).
In this module we use the vector calculus as a tool to understand some basic theories in theoretical physics. We also introduce tensors and the tensor calculus. Tensors extend the idea of a vector. A tensor is a multi-index array (e.g. a matrix) with well-defined transformation rules under coordinate transformations.
This module applies vector calculus in fluid mechanics and electromagnetism. We concentrate on fluids which do not have any resistance to flow (inviscid fluid flow) and electromagnetiism in vaccum. The mathematical models we discuss all involve solutions of equations including vector derivatives (i.e. div, grad and curl and their tensor generalisations). A particularly interesting feature of our development is the close mathematical similarity between equations from different branches of theoretical physics.