Sergei Yakovenko's blog: on Math and Teaching

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.

Exterior derivation

The differential $\mathrm df$ of a smooth function $f$ is in a sense container which conceals all directional derivatives $L_Xf=\left\langle\mathrm df,X\right\rangle$ along all directions, and dependence on $X$ is linear.

If we consider the directional Lie derivative $L_X\omega$ for a form $\omega\in\Omega^k(M)$ of degree $k\ge 1$, then simple computations show that $L_{fX}\omega$ is no longer equal to $f\cdot L_X\omega$. However, one can “correct” the Lie derivative in such a way that the result will depend on $X$ linearly. For instance, if $\omega\in\Omega^1(M)$ and $X$ is a vector field, we can define the form $\eta_X\in\Omega^1(M)$ by the identity $\eta_X=L_X\omega-\mathrm d\left\langle\omega,X\right\rangle$ and show that the 2-form $\eta(X,Y)=\left\langle\eta_X,Y\right\rangle$ is indeed bilinear antisymmetric.

The 2-form $\eta$ is called the exterior derivative of $\omega$ and denoted $\mathrm d\omega\in\Omega^2(M)$. The correspondence $\mathrm d\colon\Omega^1(M)\to\Omega^2(M)$ is an $\mathbb R$-linear operator which satisfies the Leibniz rule $\mathrm d(f\omega)=f\,\mathrm d\omega+(\mathrm df)\land \omega$ and $\mathrm d^2 f=0$ for any function $f\in\Omega^0(M)$.

It turns out that this exterior derivation can be extended to all $k$-forms preserving the above properties and is a nice (algebraically) derivation of the graded exterior algebra $\Omega^\bullet(M)=\bigoplus_{k=0}^n\Omega^k(M)$.

The lecture notes are available here.

Create a free website or blog at WordPress.com.