Maxwell’s equations

28 February 2026

Maxwell’s equations seem to be universally and consistently held up as exemplars of mathematical beauty in physical law.¹ Expressed in modern notation as differential equations, they are:

• Gauss’s law: $\nabla\cdot\vec{E}= \dfrac{\rho}{\epsilon_0}$.

• Gauss’s magnetic law: $\nabla\cdot\vec{B} = 0$.

• Maxwell–Faraday equation: $\nabla\times\vec{E} = -\dfrac{\partial\vec{B}}{\partial t}$.

• Ampère’s law: $\nabla\times\vec{B} = \mu_0 \Bigl(\vec{J} + -\epsilon_0\dfrac{\partial\vec{E}}{\partial t}\Bigr)$.

Even a reader who is unaware of the physical interpretation of the symbols can see clear symmetries in the equations.

Henri Poincaré (1854–1912) thought James Clerk Maxwell (1831–79) was able to reformulate electromagnetic theory in part due to seeing how the equations would become more symmetrical:

‘It was because Maxwell was profoundly steeped in the sense of mathematical symmetry; would he have been so, if others before him had not studied this symmetry for its own beauty?’²

But Maxwell wrote:

‘I always regarded mathematics as the method of obtaining the best shapes and dimensions of things; and this meant not only the most useful and economical, but chiefly the most harmonious and the most beautiful.’³

This does not point to using beauty in mathematics: the mathematics was a means to a beautiful end.

Beauty in mathematics is never mentioned in Maxwell’s Treatise on Electricity and Magnetism (1873). Elegance is mentioned precisely once.⁴

And Maxwell did not use the concise modern notation, and did not group the four equations together.

It was Oliver Heaviside (1850–1925) who put the equations into their modern form, which brought out their symmetry. As Heaviside’s contemporary George Francis FitzGerald (1851–1901) put it: ‘Every mathematician can appreciate the value and beauty of this.’⁵

Notes