Synopsis for Applications

Overview

In these lectures we shall consider several applications of the techniques introduced in Multivariable Calculus to problems in mathematical physics. We shall solve a range of these problems, and see how naturally the vector operators grad, div and curl arise in the expression of physical laws for gravity and electromagnetism.

Learning Outcomes

Students will consolidate their knowledge and develop their understanding of vector calculus, and learn how advances in mathematical techniques often lead to the best expression of physical laws.

Synopsis

Conservative forces and the existence of potential as a consequence of Stokes's Theorem.

Newtonian gravity and Coulomb's law as conservative forces; Gauss' theorem; potentials and Poisson's equation; boundary conditions and electrostatics.

Electric currents, Lorentz force and magnetic fields; Biot-Savart law and a derivation of Ampere's law for steady currents; the magnetostatic vector potential.

Much of the material can be found in

1. Erwin Kreyszig, Advanced Engineering Mathematics (Wiley, 8th Edition, 1999);

For the rest, consult any introductory text in electromagnetism, of which many are to be found in college libraries.