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.


Monday, December 12, 2016

Lecture 6, December 12, 2016

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.

Monday, December 5, 2016

Lecture 5, Dec 5, 2016

Multilinear antisymmetric forms and differential forms on manifolds

We discussed the module of differential 1-forms dual to the module of smooth vector fields on a manifold. Differential 1-forms are generated by differentials of smooth functions and as such can be pulled back by smooth maps.

The “raison d’être” of differential 1-forms is to be integrated over smooth curves in the manifold, the result being dependent only on the orientation of the curve and not on its specific parametrization.

At the second hour we discussed the notion of forms of higher degree, which required to introduce the Grassman algebra on the dual space T^* to an abstract finite-dimensional linear space T\simeq\mathbb R^n. The Grassmann (exterior) algebra is a mathematical miracle that was discovered by a quest for unusual and unknown, with only slight “motivations” from outside.

The day ended up with the definition of the differential k-forms and their functoriality (i.e., in what direction and how they are carried by smooth maps between manifolds).

The lecture notes are available here.

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