III.16 Differential Forms and Integration, Terence Tao: Pages 179 – 180.

A continuously differentiable k-form has a derivative that is a (k+1)-form.  An example of a the shell of a sphere and a solid sphere is given.  In the shell case the flux through the surface is used.  In the solid sphere case small compartments are used and the net flux between opposing faces is aggregated.

This may be thought of as generalizing the fundamental theorem of calculus.  This can be further generalized to yield Stokes’s theorem.  On an oriented manifold the integral of dw on the entire set S is equal to the integral of w on the oriented boundary of S.  Differentiation is said to be the adjoint of the boundary operation.

Right hand rule.

Hodge duality.

Gradient

Curl

The divergence theorem, Green’s theorem and Stokes’s theorem.

Connection between usual integrals and integration of differential forms.

“Pushes forward” and “pullback” manifolds.  Change of variables.

Pullback operation respects wedge product and the derivative.

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