# Sergei Yakovenko's blog: on Math and Teaching

## 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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