# common formulas in calculus of differential forms

## 1 Euclidean forms

To begin with we have the total differential for scalars $f\colon D\to\mathbb{R}$ where $D$ is a domain in $\mathbb{R}^{n}$:

 $df=\sum_{s}\frac{\partial f}{\partial x^{s}}dx^{s}$

or by the Einstein summation convention

 $df=\frac{\partial f}{\partial x^{s}}dx^{s}$

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

 $\nabla f=\frac{\partial f}{\partial x^{s}}$

being the components of $df$.

Here the symbols $dx^{s}$ are linear functionals $\mathbb{R}^{n}\to\mathbb{R}$ dual to the derivations $\frac{\partial}{\partial x^{s}}$, that is

 $dx^{s}\Big{(}\frac{\partial}{\partial x^{t}}\Big{)}=\delta^{s}_{t}$

this coincides with the calculation $dx^{s}(\frac{\partial}{\partial x^{t}})=\frac{\partial x^{s}}{\partial x^{t}}=% \delta^{s}_{t}$.

If $X$ is a vector field and $f$ a scalar field then one has for the directional derivative

 $Xf=X^{s}\frac{\partial}{\partial x^{s}}f=X^{s}df(\frac{\partial}{\partial x^{s% }})=df(X)$

For a pair of functions $g,f\colon D\to\mathbb{R}$ we can check Leibniz’s rule

 $d(fg)=gdf+fdg$

Let $\Omega^{0}(D)=C^{\infty}(D)$ be the set of 0-forms in $D$ and let $\Omega^{1}(D)=\{w=w_{s}dx^{s}\colon w_{s}\in\Omega^{0}\}$ (where $w_{s}dx^{s}=\sum_{s}w_{s}dx^{s}$) be the set of 1-forms in $D$.

Then the operator $d$ can be seen as a linear operator $d\colon\Omega^{0}(D)\to\Omega^{1}(D)$.

This can be generalized by defining $\Omega^{k}(D)$ to be the set of k-forms; that is, expressions of the type:

 $A_{s_{1}...s_{k}}dx^{s_{1}}\wedge\cdots\wedge dx^{s_{k}}$

where $A_{s_{1}...s_{k}}$ are in $\Omega^{0}(D)$ i.e. they are scalars and they are multi-indexed sums. Further, the symbols $dx^{s_{1}}\wedge\cdots\wedge dx^{s_{k}}$ are the wedge products of the $dx^{s}$.

So $d\colon\Omega^{k}(D)\to\Omega^{k+1}(D)$ is calculated by

 $d(A_{s_{1}...s_{k}}dx^{s_{1}}\wedge\cdots\wedge dx^{s_{k}})=d(A_{s_{1}...s_{k}% })\wedge dx^{s_{1}}\wedge\cdots\wedge dx^{s_{k}}$

For example, if $A=A_{s}dx^{s}$ then $dA=dA_{s}\wedge dx^{s}$, hence

 $dA=\frac{\partial A_{s}}{\partial x^{t}}dx^{t}\wedge dx^{s}$

which is rearranged as

 $dA=\Big{(}\frac{\partial A_{s}}{\partial x^{t}}-\frac{\partial A_{t}}{\partial x% ^{s}}\Big{)}dx^{t}\wedge dx^{s},$

and for two forms, if $B=B_{st}dx^{s}\wedge dx^{t}$ then

 $dB=\frac{\partial B_{st}}{\partial x^{u}}dx^{u}\wedge dx^{s}\wedge dx^{t}.$

Now if we have a map between two domains $F\colon D\to E$ and $F=(F^{1},...,F^{n})$, we can pullback forms as $F^{*}\colon\Omega^{k}(E)\to\Omega^{k}(D)$, beginnig with the observation that at basics $dx^{k}$, we pullback it as

 $F^{*}(dx^{k})=d(x^{k}\circ F)=dF^{k}=\frac{\partial F^{k}}{\partial x^{s}}dx^{s}$

then, if we want $\omega\mapsto F^{*}(\omega)$, where $\omega=\omega_{s_{1}...s_{k}}dx^{s_{1}}\wedge\cdots\wedge dx^{s_{k}}$, we are going to receive

 $F^{*}(\omega)=\omega_{s_{1}...s_{k}}\circ f\ \frac{\partial F^{s_{1}}}{% \partial x^{t_{1}}}\cdots\frac{\partial F^{s_{k}}}{\partial x^{t_{k}}}dx^{t_{1% }}\wedge\cdots\wedge dx^{t_{k}}$

Here the $t_{i}$-sums must be taken between all indexes obeying $1\leq t_{1}.

So if $\omega\in\Omega^{n}(D)$, $F^{*}(\omega)=\omega_{1...n}\circ F\ \det(F^{\prime})dx^{1}\wedge\cdots\wedge dx% ^{n}$

We also have

 $F^{*}(v\wedge w)=F^{*}(v)\wedge F^{*}(w)$

Obviously there are no $n+1,n+2,\ldots$ forms in $D$ and usually one set $\Omega^{k}(D)=0$ if $k\geq n$.

## 2 The de Rham complex.

The collection of mappings

 $0\longrightarrow\Omega^{0}(D)\lx@stackrel{{\scriptstyle d}}{{\longrightarrow}}% \Omega^{1}(D)\lx@stackrel{{\scriptstyle d}}{{\longrightarrow}}\cdots% \lx@stackrel{{\scriptstyle d}}{{\longrightarrow}}\Omega^{n}(D)\longrightarrow 0$

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

 $H^{k}(D)=\frac{\operatorname{ker}(d)}{\operatorname{im}(d)}$

called the cohomological $k$-group for $D$.

Some with the fear of being confused with the giving of the same name to the operator $\Omega^{k}(D)\lx@stackrel{{\scriptstyle d}}{{\longrightarrow}}\Omega^{k+1}(D)$, would like to write

 $\Omega^{k}(D)\lx@stackrel{{\scriptstyle d^{k}}}{{\longrightarrow}}\Omega^{k+1}% (D)$

and then one should modify the above conventions with

 $d^{k+1}d^{k}=0$

and

 $H^{k}(D)=\frac{\operatorname{ker}(d^{k})}{\operatorname{im}(d^{k-1})}$

## 3 Manifold’s Forms.

One had seen that for mappings $F\colon D\to E$ between $\mathbb{R}^{n}$’s domains behave as $F^{*}\colon\Omega^{k}(E)\to\Omega^{k}(D)$. Then we can assign k-forms in each chart $(U,\Phi)$ of a n-manifold $M$ by means of the coordinated functions $u^{i}=x^{i}\circ\Phi$ on the neighborhood $U$. Then

 $du^{i}=d(x^{i}\circ\Phi)=\Phi^{*}dx^{i}$

which will be the duals of the derivations $\frac{\partial}{\partial u^{j}}$.

Observe that if $\Phi^{*}\colon\Omega^{0}(\phi(U))\to\Omega^{0}(U)$ then $\Phi(g)=g\circ\Phi$ is a scalar in $U$.

If $\Phi^{*}\colon\Omega^{1}(\phi(U))\to\Omega^{1}(U)$ then

 $\Phi^{*}(w_{s}dx^{s})=w_{s}\circ\Phi\Phi^{*}(dx^{s})=w_{s}\circ\Phi du^{s}$

For $k$-forms

 $w_{s_{1}s_{2}...s_{k}}du^{s_{1}}\wedge\cdots\wedge du^{s_{k}}=w_{s_{1}s_{2}...% s_{k}}\circ\Phi^{-1}\circ\Phi d(x^{s_{1}}\circ\Phi)\wedge\cdots\wedge d(x^{s_{% k}}\circ\Phi)$
 $=\Phi^{*}(w_{s_{1}s_{2}...s_{k}}\circ\Phi^{-1})\Phi^{*}(dx^{s_{1}}\wedge\cdots% \wedge dx^{s_{k}})$
 $=\Phi^{*}(w_{s_{1}s_{2}...s_{k}}\circ\Phi^{-1}dx^{s_{1}}\wedge\cdots\wedge dx^% {s_{k}})$

where $w_{s_{1}s_{2}...s_{k}}\circ\Phi^{-1}dx^{s_{1}}\wedge\cdots\wedge dx^{s_{k}}$ is a $k$-form in $\Phi(U)$.

## 4 Forms and connections

A connection is a bi-linear operator $\nabla:\Gamma(TM)^{2}\to\Gamma(TM)$ where $\Gamma(TM)$ is the space of differentiable sections in the tangent bundle.

The Chistoffel symbols $\Gamma^{s}_{ij}$ are the components of $\nabla_{\partial_{i}}\partial_{j}$ through the equation

 $\nabla_{\partial_{i}}\partial_{j}=\Gamma^{s}_{ij}\partial_{s}$

where the $\partial_{s}$ are the coordinated tangent vectors.

The curvature tensor is defined as

 $R(X,Y)Z=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z$

which is a tri-linear map $\Gamma(TM)^{3}\to\Gamma(TM)$, so the Riemann-Chistoffel symbols are defined by the components ${R^{s}}_{ijk}$ of

 $R(\partial_{i},\partial_{j})\partial_{k}={R^{s}}_{ijk}\partial_{s}$

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

 $\nabla_{X}\partial_{j}={\omega^{s}}_{j}(X)\partial_{s}$

and

 $R(X,Y)\partial_{j}={\Omega^{s}}_{j}(X,Y)\partial_{s}$

these ${\omega^{s}}_{j}$ and ${\Omega^{s}}_{j}$ define a 1-form and a 2-form viewed as a sections $M\to\Omega^{1}(TM)$ and $M\to\Omega^{2}(TM)$ respectively.

Observe that $\nabla_{\partial_{k}}\partial_{j}={\omega^{s}}_{j}(\partial_{k})\partial_{s}$ which compared with $\nabla_{\partial_{k}}\partial_{j}=\Gamma^{s}_{kj}\partial_{s}$, it implies ${\omega^{s}}_{j}(\partial_{k})=\Gamma^{s}_{kj}$ and for an arbitrary vector field $X=X^{k}\partial_{k}$ (in the tangent coordinated basis)

 ${\omega^{s}}_{j}(X)=X^{k}\Gamma^{s}_{kj}$

Let $X_{1},X_{2},...,X_{n}$ be another frame field (the $\partial_{i}$ are the coordinated frame field) , i.e. a system of $n$-tangent vectors which are linearly independent in the tangent space, i.e, they span each $T_{p}M$.

Define thru

 $\nabla_{X_{i}}X_{j}=\hat{\Gamma}^{s}_{ij}X_{s}$

a an-holonomic connection coefficients

and

 $R(X_{i},X_{j})X_{k}={{\hat{R^{s}}}}_{ijk}X_{s}$

as the an-holonomic.

Remember that in the coordinated frame field $[\partial_{i},\partial_{j}]=0$, but since $\nabla_{X_{i}}X_{j}-\nabla_{X_{j}}X_{i}=[X_{i},X_{j}]$ this define the structural ”constants”

 ${c^{s}}_{ij}X_{s}=[X_{i},X_{j}]$

and the give relation

 ${c^{s}}_{ij}=\hat{\Gamma}^{s}_{ij}-\hat{\Gamma}^{s}_{ji}$

## 5 Cartan Structural Equations

The connection and the curvature forms satisfy the premiere $d\theta^{i}=-{\hat{\omega^{i}}}_{s}\wedge\theta^{s}$, where the $\theta^{i}$ are the 1-forms dual to the $X_{j}$ and the deuxieme ${\hat{\Omega^{i}}}_{j}=d{{\hat{\omega^{i}}}}_{j}+{{\hat{\omega^{i}}}}_{s}% \wedge{{\hat{\omega^{s}}}}_{j}$ where the corresponding connection forms are calculated by $\nabla_{Y}X_{j}=\hat{\omega^{s}}_{j}(Y)X_{s}$ i.e.

 $\hat{\omega^{l}}_{j}=\hat{\Gamma}^{l}_{js}\theta_{s}.$

All that fits perfectly to give

 $\hat{\Omega^{i}}_{j}={1\over 2}\hat{R^{i}}_{jkl}\theta^{k}\wedge\theta^{l}$

with $k.

This shows that the calculations of $\hat{R^{i}}_{jkl}$ are very easy objects to put into an algorithm (Debever).

Title common formulas in calculus of differential forms CommonFormulasInCalculusOfDifferentialForms 2013-03-22 15:51:28 2013-03-22 15:51:28 juanman (12619) juanman (12619) 37 juanman (12619) Topic msc 58A12 msc 58A10 Calculus TopicsOnCalculus