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

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 $\displaystyle d\gamma$ $\displaystyle=\tfrac{1}{2}\,\partial_{i}\gamma_{jk}\,dx^{i}\wedge dx^{j}\wedge dx% ^{k}$ $\displaystyle=\tfrac{1}{6}\,(\partial_{i}\gamma_{jk}+\partial_{j}\gamma_{ki}+% \partial_{k}\gamma_{ij})\,dx^{i}\wedge dx^{j}\wedge dx^{k}$ $\displaystyle=\sum_{i
Title formulas for differential forms of small valence FormulasForDifferentialFormsOfSmallValence 2013-03-22 15:13:04 2013-03-22 15:13:04 rmilson (146) rmilson (146) 10 rmilson (146) Theorem msc 58A10