Sergei Yakovenko's blog: on Math and Teaching

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.

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