formulas for differential forms of small valence

Coboundary formulas.

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

$df(u)=$$u(f),$

$d\alpha (u,v)=$$u(\alpha (v))-v(\alpha (u))-\alpha ([u,v]);$

$d\beta (u,v,w)=$$u(\beta (v,w))+v(\beta (w,u))+w(\beta (u,v))$

$\mathrm{ }-\beta ([u,v],w)-\beta ([v,w],u)-\beta ([w,u],v).$

Local coordinate formulas.

Let $f$ be a function, $v=v^i{\partial }_i$ a vector field, and $\alpha ={\alpha }_{i}d{x}^i$ and $\beta ={\beta }_{i}d{x}^i$ be 1-forms, and $\gamma =\frac{1}{2}{\gamma }_{ij}d{x}^i\wedge d{x}^j$ a $2$-form, expressed relative to a system of localcoordinates. The corresponding interior product expressions are:

${\iota }_v(\alpha )$$=v^i{\alpha }_i,$

${\iota }_v(\gamma )$$=v^i{\gamma }_{ij}d{x}^j.$

The exterior product formulas are:

$\alpha \wedge \beta$$={\alpha }_i{\beta }_{j}d{x}^i\wedge d{x}^j$

$=\frac{1}{2}({\alpha }_i{\beta }_j-{\alpha }_j{\beta }_i)d{x}^i\wedge d{x}^j$

$\alpha \wedge \gamma$$=\frac{1}{2}{\alpha }_i{\gamma }_{jk}d{x}^i\wedge d{x}^j\wedge d{x}^k$

$=\frac{1}{6}({\alpha }_i{\gamma }_{jk}+{\alpha }_j{\gamma }_{ki}+{\alpha }_k{\gamma }_{ij})d{x}^i\wedge d{x}^j\wedge d{x}^k$

The exterior derivative formulas are:

$df$$={\partial }_{i}fd{x}^i,$

$d\alpha$$={\partial }_i{\alpha }_{j}d{x}^i\wedge d{x}^j$

$=\frac{1}{2}({\partial }_i{\alpha }_j-{\partial }_j{\alpha }_i)d{x}^i\wedge d{x}^j$

$d\gamma$$=\frac{1}{2}{\partial }_i{\gamma }_{jk}d{x}^i\wedge d{x}^j\wedge d{x}^k$

$=\frac{1}{6}({\partial }_i{\gamma }_{jk}+{\partial }_j{\gamma }_{ki}+{\partial }_k{\gamma }_{ij})d{x}^i\wedge d{x}^j\wedge d{x}^k$