Vector Calculus for Electromagnetism

By Matthew Horner and Mike Ries

Between 1861 and 1862, James Clerk Maxwell published a set of twenty equations describing electric and magnetic fields that revolutionised the world. Oliver Heaviside then condensed these equations into the four we have today. These equations utilise vector calculus, the branch of mathematics to do with integrating and differentiating vector fields. Other areas of physics use vector calculus such as fluid dynamics and the study of gravitational fields.

Vector calculus revolves around the differential operator \( \vec{\nabla} \) which is called "del" or "nabla". It was Maxwell's "humorous" idea to name it nabla as it the word for an Egyptian harp.

Maxwell's equations in differential form

Gauss' Law for electricity:	\( \vec{\nabla} \circ \vec{E} = \frac{\rho}{\varepsilon_0} \)
Gauss' Law for magnetism:	\( \vec{\nabla} \circ \vec{B} = 0 \)
Faraday's Law:	\( \vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \)
Ampere-Maxwell Law:	\( \vec{\nabla} \times \vec{B} = \mu_0 \big(\vec{J} + \varepsilon_0 \frac{\partial \vec{E}}{\partial t}\big) \)

James Clerk Maxwell with his wife Katherine, 1869.

So let's delve into the nabla...