Vector analysis and the vector form of the field equations

1884-1885 Light and sky method

The modern vector language in which the four familiar equations of the electromagnetic field are written today. Maxwell himself had written the field relations mostly in Cartesian components, scattered across many articles of the Treatise; he gave quaternion expressions only as a separate summary he barely used.

In 1884-1885 the Englishman Oliver Heaviside recast Maxwell's theory in vector notation, threw out the potentials and kept only the fields E and H, and four recognisable equations are already visible in his work, although he never gathered them into a single group of four and reserved the name 'duplex' for the coupled pair of curl formulas. Heaviside built his vector calculus himself, at the same time as the American Josiah Willard Gibbs, who reached a similar system earlier but published later. The German Heinrich Hertz arrived independently at the same duplex form and himself conceded priority to Heaviside, which is why people long spoke of the Maxwell-Hertz equations, as Einstein still does in 1905.

On Arthur's reading, it was only the Dutchman Hendrik Lorentz in 1902 who assembled the four into one fundamental set in the form familiar today. That is, the content of the equations is Maxwell's, the vector language is Heaviside's, and the canonical set of four took shape later still.

Why this date

The year is counted from the series 'Electromagnetic Induction and Its Propagation', which Heaviside began publishing in the London journal The Electrician in 1885; by his own account he had been using his vector algebra in papers from 1883 onward, so the period is given as 1884-1885. Sources differ on how the equations are counted: Hunt assigns twenty equations to the 1873 Treatise, Rautio speaks of twenty equations in twenty variables without tying the count to a particular text, Wheeler describes the Treatise as relations scattered over more than twenty pages across two chapters, and Donaghy-Spargo counts twenty-two and moreover calls the original notation quaternionic, unlike Crowe, Hunt and Wheeler.

What it grew from

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