Maxwell's equations revisited -- mental imagery and mathematical symbols

Abstract

Using Maxwell's mental imagery of a tube of fluid motion of an imaginary fluid, we derive his equations $\mathrm{curl}\mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$, $\mathrm{curl}\mathbf{H}=\frac{\partial \mathbf{D}}{\partial t}+\mathbf{j}$, $\mathrm{div}\mathbf{D}=\varrho$, $\mathrm{div}\mathbf{B}=0$, which together with the constituting relations $\mathbf{D}={\epsilon }_0\mathbf{E}$, $\mathbf{B}={\mu }_0\mathbf{H}$, form what we call today Maxwell's equations. Main tools are the divergence, curl and gradient integration theorems and a version of Poincare's lemma formulated in vector calculus notation. Remarks on the history of the development of electrodynamic theory, quotations and references to original and secondary literature complement the paper.