formulas for differential forms of small valence

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

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

Local coordinate formulas.

Let $f$ be a function, $v=v^{i}\,\partial_{i}$ a vector field, and $\alpha=\alpha_{i}\,dx^{i}$ and $\beta=\beta_{i}\,dx^{i}$ 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:

	$\displaystyle\alpha\wedge\beta$	$\displaystyle=\alpha_{i}\beta_{j}\,dx^{i}\wedge dx^{j}$
		$\displaystyle=\tfrac{1}{2}(\alpha_{i}\beta_{j}-\alpha_{j}\beta_{i})\,dx^{i}% \wedge dx^{j}$
		$\displaystyle=\sum_{i<j}(\alpha_{i}\beta_{j}-\alpha_{j}\beta_{i})\,dx^{i}% \wedge dx^{j};$
	$\displaystyle\alpha\wedge\gamma$	$\displaystyle=\tfrac{1}{2}\,\alpha_{i}\gamma_{jk}\,dx^{i}\wedge dx^{j}\wedge dx% ^{k}$
		$\displaystyle=\tfrac{1}{6}(\alpha_{i}\gamma_{jk}+\alpha_{j}\gamma_{ki}+\alpha_% {k}\gamma_{ij})\,dx^{i}\wedge dx^{j}\wedge dx^{k}$
		$\displaystyle=\sum_{i<j<k}(\alpha_{i}\gamma_{jk}+\alpha_{j}\gamma_{ki}+\alpha_% {k}\gamma_{ij})\,dx^{i}\wedge dx^{j}\wedge dx^{k}.$

The exterior derivative formulas are:

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

Title	formulas for differential forms of small valence
Canonical name	FormulasForDifferentialFormsOfSmallValence
Date of creation	2013-03-22 15:13:04
Last modified on	2013-03-22 15:13:04
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	10
Author	rmilson (146)
Entry type	Theorem
Classification	msc 58A10