coboundary definition of exterior derivative
Let $M$ be a smooth manifold^{}, and

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let ${C}^{\mathrm{\infty}}(M)$ denote the algebra^{} of smooth functions^{} on $M$;

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let $V(M)$ denote the Liealgebra of smooth vector fields;

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and let ${\mathrm{\Omega}}^{k}(M)$ denote the vector space^{} of smooth, differential^{} $k$forms.
Recall that a differential form^{} $\alpha \in {\mathrm{\Omega}}^{k}(M)$ is a multilinear, alternating^{} mapping
$$\alpha :V(M)\times \mathrm{\cdots}\times V(M)\text{(k times)}\to {C}^{\mathrm{\infty}}(M)$$ 
such that, in local coordinates, $\alpha $ looks like a multilinear combination of its vector field arguments. Thus, employing the Einstein summation convention and local coordinates , we have
$$\alpha (u,v,\mathrm{\dots},w)={\alpha}_{ij\mathrm{\dots}k}{u}^{i}{v}^{j}\mathrm{\cdots}{w}^{k},$$ 
where $u,v,\mathrm{\dots},w$ is a list of $k$ vector fields. Recall also that ${C}^{\mathrm{\infty}}(M)$ is a $V(M)$ module. The action is given by a directional derivative^{}, and takes the form
$$v(f)={v}^{i}{\partial}_{i}f,v\in V(M),f\in {C}^{\mathrm{\infty}}(M).$$ 
With these preliminaries out of the way, we have the following description of the exterior derivative operator $d:{\mathrm{\Omega}}^{k}(M)\to {\mathrm{\Omega}}^{k+1}(M)$. For $\omega \in {\mathrm{\Omega}}^{k}(M)$, we have
$(d\omega )({v}_{0},{v}_{1},\mathrm{\dots},{v}_{k})=$  $\sum _{0\le i\le k}}{(1)}^{k}{v}_{i}\omega (\mathrm{\dots},{\widehat{v}}_{i},\mathrm{\dots})+$  (1)  
$$ 
where ${\widehat{v}}_{i}$ indicates the omission of the argument ${v}_{i}$.
The above expression (1) of $d\omega $ can be taken as the definition of the exterior derivative. Letting the ${v}_{i}$ arguments be coordinate vector fields, it is not hard to show that the above definition is equivalent to the usual definition of $d$ as a derivation of the exterior algebra of differential forms, or the local coordinate definition of $d$. The nice feature of (1) is that it is equivalent to the definition of the coboundary operator for Lie algebra cohomology. Thus, we see that de Rham cohomology^{}, which is the cohomology of the cochain complex^{} $d:{\mathrm{\Omega}}^{k}(M)\to {\mathrm{\Omega}}^{k+1}(M)$, is just zerothorder Lie algebra cohomology of $V(M)$ with coefficients in ${C}^{\mathrm{\infty}}(M)$. The bit about “zeroth order” means that we are considering cochains that are zeroth order differential operators^{} of their arguments — in other words, differential forms.
Title  coboundary definition of exterior derivative 

Canonical name  CoboundaryDefinitionOfExteriorDerivative 
Date of creation  20130322 15:38:06 
Last modified on  20130322 15:38:06 
Owner  rmilson (146) 
Last modified by  rmilson (146) 
Numerical id  15 
Author  rmilson (146) 
Entry type  Definition 
Classification  msc 15A69 
Classification  msc 58A10 
Related topic  LieAlgebraCohomology 