[KayEyeDoubleDee]

Now, I'm curious about the history of something like Maxwell's Equations vs Stokes' Theorem (or Green's Theorem, if that special case was discovered first)

[KayEyeDoubleDee]

Now, I'm curious about the history of something like Maxwell's Equations vs Stokes' Theorem (or Green's Theorem, if that special case was discovered first)

Slick, do you have any insight (or reference literature) here?

[tq]So how come we first "invent" an area of mathematics and only subsequently "discover" that it describers Nature?

[mud]You are correct sir, e.g., tensor analysis for general relativity, and functional analysis for quantum mechanics.

If you go back through a few posts you'll see that TopQuark, in particular, would argue (i think) that most physical phenomena are explained in terms of existing mathematics. I naively suggested he might have it backwards, that usually the mathematics are constructed to explain the physical process, but I think he may have a point. The example of Newton and his version of calculus clouded my vision at first...

[MUDDOG]

Both Green's and Stokes' Theorem preceded Maxwell's field equations.

Green's Theorem is the simplest extension to more than one dimension of Newton's fundamental theorem of calculus, e.g., the value of an integral can be computed by calculating the antiderivative at the boundary. In Green's case, the value of a 2-dimensional integral in 2-space equals a certain line integral about the boundary.

The classical Stokes' Theorem equates a surface integral in 3-space to a line integral on the boundary.

These classical theorems have been generalized to a modern Stokes' theorem which equates n-dimensional integrals to (n-1)-dimensional integrals on the boundary.