Sunday, July 20, 2014

Flux and the Fundamental Theorem of Calculus

Mario Rabinowitz writes, :

"Although traditionally treated the same mathematically, n dimensions in a higher dimensional space, are not the same  as strictly n-space because flux can’t be confined to just the given n dimensions in a higher space."

I agree with the importance  of the flux concept. Flux plays a fundamental role in applied mathematics. Geometers have codified the concept of Differential Form. Given an n-space, there is associated an n+1 space. I can illustrate this with two examples.

In one dimension, we have the boundary, i.e. two points of dimension zero at the ends of the line. The Fundamental Theorem of Calculus, states that the sum of line elementary sizes, is equal to the total size of the line. Every line element has two ends, when one adds them up, they cancel in pairs, leaving only the first end of the first element, and the last end of the last element. This boundary can be said to represent a flux, coming out of the line, which would be the source.

For a two-dimensional surface, we also have a lower dimensional boundary. If one adds the areas of the elements, the elementary boundaries will again cancel out. We are left with the closed line which is the boundary of the surface.

So on, and so forth.

In Field Theory, Michael Faraday, had the physical visualization of the electric and magnetic forces, well conceptualized, and James Clerk Maxwell  invented the mathematics that goes with them.

These two men represent one of the Golden Ages of England.

Then came Carl Friedrich Gauss, and his student Bernhard  Riemann, from Germany, and the French Mathematician, Élie Cartan, to finish the story of Differential Forms.