Sergei Yakovenko's blog: on Math and Teaching

Wednesday, December 21, 2016

Lecture 7, Dec 19, 2016

Integration of differential forms and the general Stokes theorem

We defined integrals of differential k-forms over certain simple geometric objects (oriented cells, smooth images of an oriented cube [0,1]^k), and extended the notion of the integral to integer combinations of cells, finite sums \sigma=\sum c_i\sigma_i,\ c_i\in\mathbb Z, so that \langle \omega,\sigma\rangle=\displaystyle\int_\sigma \omega=\sum_i c_i\int_{\sigma_i}\omega=\sum c_i \langle \omega,\sigma_i\rangle. Such combinations are called k-chains and denoted C^k(M).

Then the notion of a boundary was introduced, first for the cube, then for cells and ultimately for all chains by linearity. The property \partial\partial\sigma=0 was derived from topological considerations.

The “alternative” external derivative D on the forms was introduced as the operation conjugate to \partial so that \langle D\omega,\sigma\rangle=\langle\omega, \partial \sigma\rangle for any chain \sigma with respect to the pairing \Omega^k(M)\times C^k(M)\to \mathbb R defined by the integration. A relatively simple straightforward computation shows that for a (k-1)-form \omega=f(x)\,\mathrm dx_2\land\cdots\land \mathrm dx_n we have
D\omega=\displaystyle\frac{\partial f}{\partial x_1}\,\mathrm d x_1\land \cdots\land \mathrm dx_n, that is, D\omega=\mathrm d\omega. It follows than that D=\mathrm d on all forms, and hence we have the Stokes theorem \langle \mathrm d \omega, \sigma\rangle=\langle \omega,\partial\sigma\rangle.

Physical illustration for the Stokes theorem was given in \mathbb R^3 for the differential 1-form which is the work of the force vector field and for the 2-form of the flow of this vector field.

The class concluded by discussion of the global difference between closed and exact forms on manifolds as dual to that between cycles (chains without boundary) and exact boundaries and the Poincare lemma was proved for chains in star-shaped subdomains of \mathbb R^n.

There will be no lecture notes for this lecture, since the ideal exposition (which I tried to follow as close as possible) is in the book by V. I. Arnold, Mathematical methods of classical mechanics (2nd edition), Chapter 7, sections 35 and 36.


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