common formulas in calculus of differential forms

1 Euclidean forms

To begin with we have the total differentialMathworldPlanetmath for scalars f:D where D is a domain in n:


or by the Einstein summation convention


which are a special case of the so-called Euclidean 1-forms. Here we reconize the covariant form of the gradientMathworldPlanetmath of f in contravaiant ”state”:


being the componentsPlanetmathPlanetmath of df.

Here the symbols dxs are linear functionalsMathworldPlanetmath n dual to the derivations xs, that is


this coincides with the calculation dxs(xt)=xsxt=δts.

If X is a vector fieldMathworldPlanetmath and f a scalar field then one has for the directional derivativeMathworldPlanetmathPlanetmath


For a pair of functions g,f:D we can check Leibniz’s rule


Let Ω0(D)=C(D) be the set of 0-forms in D and let Ω1(D)={w=wsdxs:wsΩ0} (where wsdxs=swsdxs) be the set of 1-forms in D.

Then the operator d can be seen as a linear operator d:Ω0(D)Ω1(D).

This can be generalized by defining Ωk(D) to be the set of k-forms; that is, expressions of the type:


where As1sk are in Ω0(D) i.e. they are scalars and they are multi-indexed sums. Further, the symbols dxs1dxsk are the wedge productsPlanetmathPlanetmath of the dxs.

So d:Ωk(D)Ωk+1(D) is calculated by


For example, if A=Asdxs then dA=dAsdxs, hence


which is rearranged as


and for two forms, if B=Bstdxsdxt then


Now if we have a map between two domains F:DE and F=(F1,,Fn), we can pullback forms as F*:Ωk(E)Ωk(D), beginnig with the observation that at basics dxk, we pullback it as


then, if we want ωF*(ω), where ω=ωs1skdxs1dxsk, we are going to receive


Here the ti-sums must be taken between all indexes obeying 1t1<t2<<tkn.

So if ωΩn(D), F*(ω)=ω1nFdet(F)dx1dxn

We also have


Obviously there are no n+1,n+2, forms in D and usually one set Ωk(D)=0 if kn.

2 The de Rham complex.

The collection of mappings


give us a chain complex due that dd=0, so one can measure how much this differs from exactness via its homology


called the cohomological k-group for D.

Some with the fear of being confused with the giving of the same name to the operator Ωk(D)dΩk+1(D), would like to write


and then one should modify the above conventions with




3 Manifold’s Forms.

One had seen that for mappings F:DE between n’s domains behave as F*:Ωk(E)Ωk(D). Then we can assign k-forms in each chart (U,Φ) of a n-manifold M by means of the coordinated functions ui=xiΦ on the neighborhood U. Then


which will be the duals of the derivations uj.

Observe that if Φ*:Ω0(ϕ(U))Ω0(U) then Φ(g)=gΦ is a scalar in U.

If Φ*:Ω1(ϕ(U))Ω1(U) then


For k-forms


where ws1s2skΦ-1dxs1dxsk is a k-form in Φ(U).

4 Forms and connections

A connectionMathworldPlanetmath is a bi-linear operator :Γ(TM)2Γ(TM) where Γ(TM) is the space of differentiableMathworldPlanetmath sections in the tangent bundle.

The Chistoffel symbols Γijs are the components of ij through the equation


where the s are the coordinated tangent vectorsMathworldPlanetmath.

The curvature tensor is defined as


which is a tri-linear map Γ(TM)3Γ(TM), so the Riemann-Chistoffel symbols are defined by the components Rsijk of


With these one define the connection forms and the curvature forms as




these ωsj and Ωsj define a 1-form and a 2-form viewed as a sections MΩ1(TM) and MΩ2(TM) respectively.

Observe that kj=ωsj(k)s which compared with kj=Γkjss, it implies ωsj(k)=Γkjs and for an arbitrary vector field X=Xkk (in the tangentPlanetmathPlanetmath coordinated basis)


Let X1,X2,,Xn be another frame field (the i are the coordinated frame field) , i.e. a system of n-tangent vectors which are linearly independentMathworldPlanetmath in the tangent space, i.e, they span each TpM.

Define thru


a an-holonomic connection coefficients



as the an-holonomic.

Remember that in the coordinated frame field [i,j]=0, but since XiXj-XjXi=[Xi,Xj] this define the structural ”constants”


and the give relation


5 Cartan Structural Equations

The connection and the curvature forms satisfy the premiere dθi=-ωi^sθs, where the θi are the 1-forms dual to the Xj and the deuxieme Ωi^j=dωi^j+ωi^sωs^j where the corresponding connection forms are calculated by YXj=ωs^j(Y)Xs i.e.


All that fits perfectly to give


with k<l.

This shows that the calculations of Ri^jkl are very easy objects to put into an algorithm (Debever).

Title common formulas in calculus of differential forms
Canonical name CommonFormulasInCalculusOfDifferentialForms
Date of creation 2013-03-22 15:51:28
Last modified on 2013-03-22 15:51:28
Owner juanman (12619)
Last modified by juanman (12619)
Numerical id 37
Author juanman (12619)
Entry type Topic
Classification msc 58A12
Classification msc 58A10
Related topic Calculus
Related topic TopicsOnCalculus