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Did he invent surface and line integrals, or did they already exist when he formulated his equations. If they did, already exist, how did they come about in pure math?

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Maxwell was not solely responsible for vector calculus, the formalism was developed before his treatise by Gibbs and Heaviside, as well as others. – JamalS Apr 26 '14 at 21:42
Also, if you read his treatise, most of the modern notation is absent. No $\nabla \times \vec{E}$ type stuff, so it's a lot more verbose. – webb Apr 26 '14 at 22:26
Wasn't Ampère cast Farady's results in mathematical form. Poisson was a real mathematical physicist, too. – Geremia Apr 28 '14 at 2:54
up vote 2 down vote accepted

No, Maxwell did not invent surface or line integrals. See History of Stokes' Theorem, which explains surface integrals were in use earlier.

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