When Maxwell first derived his equations in 1861, he did it without a cross product. As a matter of fact, the vector analysis of Gibbs and Heaviside was develop

When Maxwell first derived his equations in 1861, he did it without a cross product. As a matter of fact, the vector analysis of Gibbs and Heaviside was developed to formulate Maxwell equations. The most elegant formulation of Maxwell equations is obtained, in my opinion, using Geometric Algebra, which reduces them to a single and very simple equation.