PlanetMath: formulas for differential forms of small valence

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formulas for differential forms of small valence

(Theorem)

Coboundary formulas.

Given a function $f$ (same thing as a differential $0$ -form), a differential 1-form $\alpha$ and a differential 2-form $\beta$ , and for vector fields $u,v,w$ , we have

$\displaystyle d f(u) =$	$\displaystyle u(f),$
$\displaystyle d \alpha(u,v) =$	$\displaystyle u(\alpha(v)) - v(\alpha(u))- \alpha([u,v]);$
$\displaystyle d \beta(u,v,w) =$	$\displaystyle u( \beta(v,w)) +v(\beta(w,u)) + w(\beta(u,v))$
	$\displaystyle \ -\beta([u,v],w) - \beta([v,w],u) - \beta([w,u],v).$

Let $f$ be a function, $v=v^i\, \partial_i$ a vector field, and $\alpha = \alpha_i\, d x^i$ and $\beta = \beta_i\, d x^i$ be 1-forms, and $\gamma=\tfrac{1}{2} \gamma_{ij}\, dx^i \wedge dx^j$ a $2$ -form, expressed relative to a system of local coordinates. The corresponding interior product expressions are:

$\displaystyle \iota_v(\alpha)$	$\displaystyle = v^i \alpha_i ,$
$\displaystyle \iota_v(\gamma)$	$\displaystyle = v^i \gamma_{ij}\, dx^j.$

The exterior product formulas are:

$\displaystyle \alpha\wedge\beta$	$\displaystyle = \alpha_i \beta_j\, dx^i\wedge dx^j$
	$\displaystyle =\tfrac{1}{2} (\alpha_i \beta_j - \alpha_j \beta_i)\, d x^i \wedge d x^j$
	$\displaystyle = \sum_{i<j} (\alpha_i \beta_j - \alpha_j \beta_i)\, d x^i \wedge d x^j;$
$\displaystyle \alpha\wedge\gamma$	$\displaystyle = \tfrac{1}{2} \, \alpha_i \gamma_{jk}\, dx^i\wedge dx^j\wedge dx^k$
	$\displaystyle =\tfrac{1}{6} (\alpha_i \gamma_{jk} + \alpha_j \gamma_{ki} + \alpha_k \gamma_{ij})\, d x^i \wedge d x^j \wedge d x^k$
	$\displaystyle =\sum_{i<j<k} (\alpha_i \gamma_{jk} + \alpha_j \gamma_{ki} + \alpha_k \gamma_{ij})\, d x^i \wedge d x^j \wedge d x^k.$

The exterior derivative formulas are:

$\displaystyle df$	$\displaystyle = \partial_i f\, dx^i,$
$\displaystyle d\alpha$	$\displaystyle = \partial_i \alpha_j \,dx^i\wedge dx^j$
	$\displaystyle =\tfrac{1}{2}\,(\partial_i \alpha_j-\partial_j\alpha_i) \,dx^i\wedge dx^j$
	$\displaystyle =\sum_{i<j} (\partial_i \alpha_j-\partial_j\alpha_i) \,dx^i\wedge dx^j;$
$\displaystyle d\gamma$	$\displaystyle = \tfrac{1}{2}\, \partial_i \gamma_{jk} \,dx^i\wedge dx^j\wedge dx^k$
	$\displaystyle =\tfrac{1}{6}\,(\partial_i \gamma_{jk}+\partial_j \gamma_{ki} +\partial_k \gamma_{ij}) \,dx^i\wedge dx^j\wedge dx^k$
	$\displaystyle =\sum_{i<j<k}(\partial_i \gamma_{jk}+\partial_j \gamma_{ki} +\partial_k \gamma_{ij}) \,dx^i\wedge dx^j\wedge dx^k .$