formulas for differential forms of small valence

Coboundary formulas.

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

df(u)= u(f),
dα(u,v)= u(α(v))-v(α(u))-α([u,v]);
dβ(u,v,w)= u(β(v,w))+v(β(w,u))+w(β(u,v))

Local coordinate formulas.

Let f be a function, v=vii a vector field, and α=αidxi and β=βidxi be 1-forms, and γ=12γijdxidxj a 2-form, expressed relative to a system of local coordinates. The corresponding interior product expressions are:

ιv(α) =viαi,
ιv(γ) =viγijdxj.

The exterior product formulas are:

αβ =αiβjdxidxj
αγ =12αiγjkdxidxjdxk

The exterior derivative formulas are:

df =ifdxi,
dα =iαjdxidxj
dγ =12iγjkdxidxjdxk
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