Exterior Algebra

Let \(V\) be a real vector space of dimension \(n\).

Definition, Alternating algebraic forms:

For each \(k\), we define \(\Alt^k V\) as the space of alternating \(k\)-linear maps \(V \times \cdots \times V \rightarrow \mathbb{R}\).

Note

• \(\Alt^0 = \mathbb{R}\),

• \(\Alt^1 = V^{*}\) is the dual space of \(V\) (the space of covectors)

Definition, Exterior product:

For \(\omega \in \Alt^j\) and \(\eta \in \Alt^k\), their exterior (wedge) product is given by:

\[(\omega \wedge \eta ) (v_1, \cdots, v_{j+k}) = \sum_{\sigma} (\mathrm{sign}~ \sigma) \omega (v_{\sigma(1)}, \cdots, v_{\sigma(j)}) \eta (v_{\sigma(j+1)}, \cdots, v_{\sigma(j+k)})\]

for all \(v_i \in V\). Where the sum is over all permutations \(\sigma\) of \(\{ 1,\cdots,j+k \}\), for which \(\sigma(1)< \cdots <\sigma(j)\) and \(\sigma(j+1)< \cdots <\sigma(j+k)\).

Note

• The exterior product is bilinear, associative,

• anti-commutative: \(\eta \wedge \omega = (-1)^{jk} \omega \wedge \eta\) for all \(\omega \in \Alt^j\) and \(\eta \in \Alt^k\).

Definition, Grassmann Algebra:

Grassmann Algebra is defined by:

\[\Alt V := \bigoplus_k \Alt^k V\]

This is a anti-commutative graded algebra. Also called Exterior Algebra of \(V^{*}\)

In the case of \(V=\mathbb{R}^n\), we have:

• \(\Alt V^0 \sim \mathbb{R}\),

• \(\Alt V^1 \sim \mathbb{R}^n\),

• \(\Alt V^{n-1} \sim \mathbb{R}^n\), using Riesz representation theorem,

• \(\Alt V^n \sim \mathbb{R}\), using the map \(v \longmapsto \det(v,v_1,\cdots,v_{n-1})\).

Basis

Let \(v_1,\cdots,v_n\) be a basis of \(V\) and \(\mu_1,\cdots,\mu_n\) the associated dual basis for \(V^*\) (\(\mu_i(v_j) = \delta_{ij}\)).

For any increasing permutations \(\sigma, \rho : \{ 1,\cdots,k \} \longrightarrow \{ 1,\cdots,n \}\), we have:

\[\mu_{\sigma(1)} \wedge \cdots \wedge \mu_{\sigma(k)} (v_{\rho(1)}, \cdots, v_{\rho(k)}) = \chi_{\sigma,\rho}\]

thus the \(\binom {n}{k}\) algebraic \(k\)-forms \(\mu_{\sigma(1)} \wedge \cdots \wedge \mu_{\sigma(k)}\), form a basis for \(\Alt^k V\) and \(\dim \Alt^k V = \binom {n}{k}\).

Definition, Interior product:

Let \(\omega\) be a \(k\)-form, and \(v \in V\). The interior product of \(\omega\) and \(v\) is the \((k-1)\)-form \(\omega \lrcorner v\) defined by:

\[\omega \lrcorner v (v_1,\cdots,v_{k-1}) = \omega (v,v_1,\cdots,v_{k-1})\]

• We have for \(\omega \in \Alt^k V\), \(\eta \in \Alt^l V\) and \(v \in V\):

\[(\omega \wedge \eta) \lrcorner v = (\omega \lrcorner v)\wedge \eta + (-1)^k \omega \wedge (\eta \lrcorner v)\]

Definition, Inner product:

If \(V\) is has an inner product, then \(\Alt^k V\) is endowed with an inner product given by:

\[(\omega , \eta) = \sum_{\rho} \omega (v_{\rho(1)}, \cdots, v_{\rho(k)}) \eta (v_{\rho(1)}, \cdots, v_{\rho(k)}), ~~~\forall \omega, \eta \in \Alt^k V.\]

where the sum is over increasing sequences \(\rho : \{ 1,\cdots,k \} \longrightarrow \{ 1,\cdots,n \}\), and \(v_1, \cdots,v_n\) is any orthonormal basis.

Orientation and Volume form

Todo

add Orientation and Volume form

Definition, Pullback:

A linear transformation of vector spaces \(L: V \rightarrow W\) induces a transformation \(L^{*}: \Alt W \rightarrow \Alt V\), called the pullback, and given by:

\[L^{*} \omega (v_{1}, \cdots, v_{k}) = \omega (L v_{1}, \cdots, L v_{k}),~~~~~ \forall \omega \in \Alt^k W,~~~ v_{1}, \cdots, v_{k} \in V\]

• The pullback acts contravariantly: if \(U \xrightarrow{~K~} V \xrightarrow{~L~} W\) then,

  \[\Alt W \xrightarrow{~K^{*}~} \Alt V \xrightarrow{~L^{*}~} \Alt U\]

• \(L^{*} (\omega \wedge \eta) = L^{*} \omega \wedge L^{*} \eta\)

Let V be a subspace of W. For the inclusion \(\imath_V : V \longrightarrow W\), we can define its pullback \(\imath_V^{*}\): this is a surjection of \(\Alt W\) onto \(\Alt V\).

If W has an inner product and \(\pi_V : W \longrightarrow V\) is the orthogonal projection. We can define its pullback \(\pi_V^{*}\) : this an injection of \(\Alt V\) onto \(\Alt W\).

Let us consider the composition : \(W\) shortstack{\(\pi_V\) \ \(\longrightarrow\)} \(V\) shortstack{\(\imath_V\) \ \(\longrightarrow\)} \(W\), and its pullback \(\pi_V^* \imath_V^*\).

Definition, The tangential and normal parts:

• \(\pi_V^* \imath_V^*\) associates for each \(\omega \in \Alt^k\) its tangential part \(\omega_{\parallel}\) with respect to \(V\) :

\[(\pi_V^* \imath_V^* \omega) (v_1,\cdots,v_k) = \omega (\pi_V v_1, \cdots, \pi_V v_k), ~~~~~\forall v_1,\cdots,v_k \in W.\]

• \(\omega - \pi_V^* \imath_V^* \omega\) associates for each \(\omega \in \Alt^k\) its normal part \(\omega_{\perp}\) with respect to \(V\).

The tangential part of \(\omega\) vanishes if and only if the image of \(\omega\) in \(\Alt^k V\) vanishes.

Let \(V\) be an oriented inner product space, with volume form \(\mbox{vol}\). Let \(\omega \in \Alt^k V\). We can define a new linear map \(L_{\omega}\) as the composition of \(\Alt^{n-k} V \longrightarrow \Alt^n V\) such as:

\[\mu \longmapsto \omega \wedge \mu\]

and the canonical isomorphism of \(\Alt^n V\) onto \(\mathbb{R}\), and using the Riesz representation theorem, there exists an element \(\star \omega \in \Alt^{n-k} V\) such that : \(L_{\omega} (\mu) = (\star \omega , \mu)\), i.e.:

\[\omega \wedge \mu = (\star \omega , \mu) \mbox{vol}, ~~~\omega \in \Alt^{k}, ~\mu \in \Alt^{n-k}\]

Definition, The Hodge star operation:

The linear map which maps \(\Alt^k V\) onto \(\Alt^{n-k} V\) \(\omega \longmapsto \star \omega\) is called the Hodge star operator.

• If \(e_1,\cdots,e_n\) is any positively oriented orthonormal basis, and \(\sigma\) a permutation, we have

\[\omega(e_{\sigma(1)}, \cdots, e_{\sigma(k)}) = (\mathrm{sign} \sigma) \star \omega(e_{\sigma(k+1)}, \cdots, e_{\sigma(n)})\]

• \(\star \star \omega = (-1)^{k(n-k)} \omega, ~~~\forall \omega \in \Alt^k V\), thus the Hodge star is an isometry.

• \((\star \omega)_{\parallel} = \star (\omega_{\perp})\) and \((\star \omega)_{\perp} = \star (\omega_{\parallel})\)

• the image of \(\star \omega\) in \(\Alt^k V\) vanishes if and only if \(\omega_{\perp}\) vanishes.

\[\begin{split}\begin{tabular}{|c|l|} \hline $\Alt^0 \mathbb{R}^3 \cong \mathbb{R}$ & $c \leftrightarrow c$ \\ % \hline $\Alt^1 \mathbb{R}^3 \cong \mathbb{R}^3$ & $u_1 \diff x_1 + u_2 \diff x_2 + u_3 \diff x_3 \leftrightarrow u$ \\ % \hline $\Alt^2 \mathbb{R}^3 \cong \mathbb{R}^3$ & $u_3 \diff x_1 \wedge \diff x_2 - u_2 \diff x_1 \wedge \diff x_3 + u_1 \diff x_2 \wedge \diff x_3 + \leftrightarrow u$ \\ % \hline $\Alt^3 \mathbb{R}^3 \cong \mathbb{R}$ & $c \diff x_1 \wedge \diff x_2 \wedge \diff x_3 \leftrightarrow c$ \\ \hline \end{tabular}\end{split}\]

\[\begin{split}\begin{tabular}{|c|l|} \hline $ \wedge : \Alt^1 \mathbb{R}^3 \times \Alt^1 \mathbb{R}^3 \longrightarrow \Alt^2 \mathbb{R}^3$ & $\times : \mathbb{R}^3 \times \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ \\ $ \wedge : \Alt^1 \mathbb{R}^3 \times \Alt^2 \mathbb{R}^3 \longrightarrow \Alt^3 \mathbb{R}^3$ & $\cdot : \mathbb{R}^3 \times \mathbb{R}^3 \longrightarrow \mathbb{R}$ \\ \hline \end{tabular}\end{split}\]

\[\begin{split}\begin{tabular}{|c|l|} \hline $ L^* : \Alt^0 \mathbb{R}^3 \longrightarrow \Alt^0 \mathbb{R}^3 $ & $\id : \mathbb{R} \longrightarrow \mathbb{R}$ \\ $ L^* : \Alt^1 \mathbb{R}^3 \longrightarrow \Alt^1 \mathbb{R}^3 $ & $L^T : \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ \\ $ L^* : \Alt^2 \mathbb{R}^3 \longrightarrow \Alt^2 \mathbb{R}^3 $ & $(\det L )L^{-1} : \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ \\ $ L^* : \Alt^3 \mathbb{R}^3 \longrightarrow \Alt^3 \mathbb{R}^3 $ & $(\det L) : \mathbb{R} \longrightarrow \mathbb{R}$ ~~~($c \longmapsto c \det L$) \\ \hline \end{tabular}\end{split}\]

\[\begin{split}\begin{tabular}{|c|l|} \hline $ \lrcorner v : \Alt^1 \mathbb{R}^3 \longrightarrow \Alt^0 \mathbb{R}^3 $ & $v \cdot : \mathbb{R}^3 \longrightarrow \mathbb{R}$ \\ $ \lrcorner v : \Alt^2 \mathbb{R}^3 \longrightarrow \Alt^1 \mathbb{R}^3 $ & $v \times : \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ \\ $ \lrcorner v : \Alt^3 \mathbb{R}^3 \longrightarrow \Alt^2 \mathbb{R}^3 $ & $v : \mathbb{R} \longrightarrow \mathbb{R}^3$ ~~~($c \longmapsto c v$) \\ \hline \end{tabular}\end{split}\]

\[\begin{split}\begin{tabular}{|c|l|} \hline inner product on $\Alt^k \mathbb{R}^3$ induced & dot product on $\mathbb{R}$ and $\mathbb{R}^3$ \\ by dot product on $\mathbb{R}^3$ & \\ $\volume = \diff x_1 \wedge \diff x_2 \wedge \diff x_3$ & $(v_1,v_2,v_3) \longmapsto \det(v_1|v_2|v_3)$ \\ \hline \end{tabular}\end{split}\]

\[\begin{split}\begin{tabular}{|c|l|} \hline $ \star : \Alt^0 \mathbb{R}^3 \longrightarrow \Alt^3 \mathbb{R}^3 $ & $\id : \mathbb{R} \longrightarrow \mathbb{R}$ \\ $ \star : \Alt^1 \mathbb{R}^3 \longrightarrow \Alt^2 \mathbb{R}^3 $ & $\id : \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ \\ \hline \end{tabular}\end{split}\]

Exterior Calculus on manifolds and Differential forms

Let \(\Omega\) be a smooth manifold, of dimension \(n\).

• \(\forall x \in \Omega\) we denote by \(T_x \Omega\) the tangent space. This is a vector space of dimension \(n\),

• tangent bundle \(\{ (x,v), ~~ x \in \Omega, v \in T_x \Omega \}\),

• Applying the exterior algebra to the tangent spaces, we obtain the exterior forms bundle, whose elements are pairs \((x,\mu)\) with \(x \in \Omega\) and \(\mu \in \Alt^k T_x \Omega\).

• a differential \(k\)-form \(\omega\) is a section of this bundle. This is a map which associates to each \(x \in \Omega\) an element \(\omega_x \in \Alt^k T_x \Omega\),

• if the map \(\mathcal{L}_{\omega}^k : x \longmapsto \omega_x (v_1(x), \cdots, v_k(x))\) is smooth (whenever \(v_i\) are smooth), we say that \(\omega\) is a smooth differential \(k\)-form,

• we define \(\Lambda^k(\Omega)\) the space of all smooth \(k\)-forms on \(\Omega\),

• \(\Lambda^0(\Omega) = \mathcal{C}^{\infty}(\Omega)\),

• if the map \(\mathcal{L}_{\omega}^k\) is \(\mathcal{C}^{m}(\Omega)\), we define differential \(k\)-forms with less smoothness \(\mathcal{C}^{m} \Lambda^k (\Omega)\).

Let \(\Omega\) be a smooth manifold, of dimension \(n\).

Exterior product:

if \(\omega \in \Lambda^k(\Omega)\) and \(\eta \in \Lambda^j(\Omega)\), we may define \(\omega \wedge \eta\) as \((\omega \wedge \eta)_x = \omega_x \wedge \eta_x\) and the Grassmann algebra \(\Lambda(\Omega) := \bigoplus_k \Lambda^k(\Omega)\)

Differential forms can be differentiated and integrated, without recourse to any additional structure, such as a metric or a measure.

Exterior differentiation:

For each \(\omega \in \Lambda^k(\Omega)\), can define the \((k+1)\)-form \(\diff \omega \in \Lambda^{k+1}(\Omega)\), such as:

\[\diff\omega_x(v_1,\cdots,v_{k+1}) = \sum_{j=1}^{k+1} (-1)^j \partial_{v_j} \omega_x(v_1,\cdots,\hat{v_j},\cdots,v_{k+1})\]

where the hat is used to indicated a suppressed argument.

This defines a graded linear operator of degree \(+1\), of \(\Lambda(\Omega)\) onto \(\Lambda(\Omega)\).

We have the following properties:

• \(\diff \circ \diff = 0\)

• \(\diff (\omega \wedge \eta) = \diff \omega \wedge \eta + (-1)^k \omega \wedge \diff \eta, ~~\forall \omega \in \Lambda^k(\Omega), \eta \in \Lambda^j(\Omega)\),

• (Pullback) let \(\phi\) be a smooth map of \(\Omega\) onto \(\Omega^{\prime}\). Then \(\phi^*(\omega \wedge \eta) = \phi^*(\omega) \wedge \phi^*(\eta)\) and \(\phi^* (\diff \omega) = \diff (\phi^* \omega)\),

• (Interior product) the interior product of a differential \(k\)-form \(\omega\) with a vector field \(v\),

• we obtain a \((k-1)\)-form by : \((\omega \lrcorner v)_x := \omega_x \lrcorner v_x\),

• (Trace operator) the pullback \(i_{\partial \Omega}^*\) of \(i_{\partial \Omega}\) is the trace operator \(\trace\)

Integration:

• If \(f\) is an oriented, piecewise smooth \(k\)-dimensional submanifold of \(\Omega\), and \(\omega\) is a continuous \(k\)-form, then th integral \(\int_f \omega\) is well defined :

  • [0-forms] can be evaluated at points,

  • [1-forms] can be integrated over directed curves,

  • [2-forms] can be integrated over directed surfaces,

• (Inner product) The \(L^2\)-inner product of two differential \(k\)-forms on an oriented Riemannian manifold \(\Omega\) is defined as :

\[(\omega,\eta)_{L^2 \Lambda^k} = \int_{\Omega} (\omega_x,\eta_x) \volume = \int \omega \wedge \star \eta\]

The completion of \(\Lambda^k(\Omega)\) in the corresponding norm defines the Hilbert space \(L^2 \Lambda^k(\Omega)\).

We have the following results:

• (Integration) if \(\phi\) is an orientation-preserving diffeomorphism, then

\[\int_{\Omega} \phi^* \omega = \int_{\Omega^{\prime}} \omega, ~~~ \forall \omega \in \Lambda^n(\Omega^{\prime})\]

Theorem, Stokes theorem:

If \(\Omega\) is an oriented \(n\)-manifold with boundary \(\partial \Omega\), then

\[\int_{\Omega} \diff \omega = \int_{\partial \Omega} \trace \omega, ~~~ \forall \omega \in \Lambda^{n-1}(\Omega)\]

Theorem, Integration by parts:

If \(\Omega\) is an oriented \(n\)-manifold with boundary \(\partial \Omega\), then

\[\int_{\Omega} \diff \omega \wedge \eta = (-1)^{k-1} \int_{\Omega} \omega \wedge \diff \eta + \int_{\partial \Omega} \trace \omega \wedge \trace \eta, ~~~ \forall \omega \in \Lambda^{k}(\Omega), \eta \in \Lambda^{n-k-1}(\Omega)\]

Sobolev spaces of differential forms

As for the classical case, we can define the Sobolev spaces as:

• \(H^s \Lambda^k(\Omega)\) is the space of differential \(k\)-forms such that \(\mathcal{L}_{\omega}^k \in H^s(\Omega)\).

• \(H \Lambda^k(\Omega) = \{ \omega \in L^2 \Lambda^k(\Omega),~~ \diff \omega \in L^2 \Lambda^{k+1}(\Omega) \}\). The associated norm is :

\[\| \omega \|_{H \Lambda^k}^2 = \| \omega \|_{H \Lambda}^2 := \| \omega \|_{L^2 \Lambda^k}^2 + \| \diff \omega \|_{L^2 \Lambda^{k+1}}^2\]

• \(H \Lambda^{0}(\Omega)\) coincides with \(H^1 \Lambda^{0}(\Omega)\),

• \(H \Lambda^{n}(\Omega)\) coincides with \(L^2 \Lambda^{n}(\Omega)\),

• for \(0 < k < n\), we have \(H^1 \Lambda^k(\Omega) \subset H \Lambda^k(\Omega) \subset L^2 \Lambda^k(\Omega)\), strictly.

\[\begin{split}\begin{tabular}{|c|c c c c c|} \hline $k$ & $\Lambda^k$ & $H \Lambda^k$ & $\diff \omega$ & $\int_f \omega$ & $\kappa \omega$ \\ \hline & & & & & \\ 0 & $\mathcal{C}^{\infty}$ & $H^1$ & $\nabla \omega$ & $\omega(f)$ & $0$ \\ 1 & $\mathcal{C}^{\infty}(\mathbb{R}^3)$ & $H(\rots,\mathbb{R}^3)$ & $\rots \omega$ & $\int_f \omega \cdot t \diff \mathcal{H}_1$ & $x \longmapsto x \cdot \omega(x)$ \\ 2 & $\mathcal{C}^{\infty}(\mathbb{R}^3)$ & $H(\divs, \mathbb{R}^3)$ & $\divs \omega$ & $\int_f \omega \cdot n \diff \mathcal{H}_2$ & $x \longmapsto x \times \omega(x)$ \\ 3 & $\mathcal{C}^{\infty}$ & $L^2$ & $0$ & $\int_f \omega \diff \mathcal{H}_3$ & $x \longmapsto x \omega(x)$ \\ & & & & & \\ \hline \end{tabular}\end{split}\]

Cohomology and De Rham Complex

The De Rham complex is the sequence of spaces and mappings

\[0 \xrightarrow{\quad} \Lambda^0(\Omega) \xrightarrow{~\diff~} \Lambda^1(\Omega) \xrightarrow{~\diff~} \cdots \xrightarrow{~\diff~} \Lambda^n(\Omega) \xrightarrow{\quad} 0\]

Since, \(\diff \circ \diff = 0\), we have

\[\mathcal{R}(\diff : \Lambda^{k-1}(\Omega) \longrightarrow \Lambda^k(\Omega)) \subset \mathcal{N}(\diff : \Lambda^{k}(\Omega) \longrightarrow \Lambda^{k+1}(\Omega))\]

If \(\Omega\) is an oriented Riemannian manifold, we have the following cohomology:

\[0 \xrightarrow{\quad} H \Lambda^0(\Omega) \xrightarrow{~\diff~} H \Lambda^1(\Omega) \xrightarrow{~\diff~} \cdots \xrightarrow{~\diff~} H \Lambda^n(\Omega) \xrightarrow{\quad} 0\]

The coderivative operator \(\delta : \Lambda^{k}(\Omega) \longrightarrow \Lambda^{k-1}(\Omega)\) is defined as:

\[\star \delta \omega = (-1)^k \diff \star \omega,~~~ \omega \in \Lambda^k(\Omega)\]

• we have

\[(\diff \omega , \eta ) = (\omega , \delta \eta ) + \int_{\partial \Omega} \trace \omega \wedge \trace \eta, ~~~ \forall \omega \in \Lambda^{k}(\Omega), \eta \in \Lambda^{k+1}(\Omega),\]

• \(\delta\) is a graded linear operator of degree \(-1\).

• \(\delta\) is the formal adjoint of \(\diff\) whenever \(\omega\) or \(\eta\) vanishes near the boundary.

• we define the spaces

\[H^* \Lambda^k(\Omega) = \{ \omega \in L^2 \Lambda^k(\Omega),~~ \delta \omega \in L^2 \Lambda^{k-1}(\Omega) \}.\]

we have \(H^* \Lambda^k(\Omega) = \star H \Lambda^{n-k}(\Omega)\).

• we obtain the dual complex

\[0 \xleftarrow{\quad} H^* \Lambda^0(\Omega) \xleftarrow{~\delta~} H^* \Lambda^1(\Omega) \xleftarrow{~\delta~} \cdots \xleftarrow{~\delta~} H^* \Lambda^n(\Omega) \xleftarrow{\quad} 0\]

Cohomology with boundary conditions

Let \(\Lambda_0^k(\Omega)\) be the subspace of \(\Lambda^k(\Omega)\) of smooth \(k\)-forms with compact support. We have \(\diff \Lambda_0^k \subset \Lambda_0^{k+1}\).

The De Rham complex with the compact support is

\[0 \xrightarrow{\quad} \Lambda^0_0(\Omega) \xrightarrow{~\diff~} \Lambda^1_0(\Omega) \xrightarrow{~\diff~} \cdots \xrightarrow{~\diff~} \Lambda^n_0(\Omega) \xrightarrow{\quad} 0\]

Recall that the closure of \(\Lambda_0^k(\Omega)\) in \(H \Lambda^k(\Omega)\) is

\[H_0 \Lambda^k(\Omega) = \{ \omega \in H \Lambda^k(\Omega),~~ \trace \omega =0\}.\]

The \(L^2\) version of the last complex is

\[0 \xrightarrow{\quad} H_0 \Lambda^0(\Omega) \xrightarrow{~\diff~} H_0 \Lambda^1(\Omega) \xrightarrow{~\diff~} \cdots \xrightarrow{~\diff~} H_0 \Lambda^n(\Omega) \xrightarrow{\quad} 0\]

Definition, Harmonic forms:

The harmonic \(k\)-forms are the differential \(k\)-forms that verify the differential equations

\[\begin{split}\left\{ \begin{aligned} \diff \omega &=& 0,\\ \delta \omega &=& 0,\\ \trace \star \omega &=& 0.\\ \end{aligned} \right.\end{split}\]

this defines the following space,

\[\mathfrak{H}^k (\Omega) = \{ \omega \in H \Lambda^k(\Omega) \cap H_0^* \Lambda^k(\Omega),~~\diff \omega = 0, \delta \omega = 0 \}\]

We can also define the following space,

\[\mathfrak{H}_0^k (\Omega) = \{ \omega \in H_0 \Lambda^k(\Omega) \cap H^* \Lambda^k(\Omega),~~\diff \omega = 0, \delta \omega = 0 \}\]

As we can see, \(\star \mathfrak{H}^k (\Omega) = \mathfrak{H}_0^{n-k} (\Omega)\).

Proposition, Poincaré duality:

There is an isomorphism between the \(k\) th De Rham cohomology space and the \((n-k)\) th cohomology space with boundary conditions.

Homological Algebra and Hilbert complexes

Homological Algebra

• A cochain complex is a sequence of vector spaces and linear maps

• \(k\)-cocycles \(\mathfrak{Z}^k := \mathcal{N}(d_k)\),

• \(k\)-coboundaries \(\mathfrak{B}^k := \mathcal{R}(d_{k-1})\),

• \(k\)-cohomology \(\mathcal{H}^k(V) := \mathfrak{Z}^k / \mathfrak{B}^k\),

• we say that the sequence is exact, if the cohomology vanishes (i.e. \(\forall~k,~~ \mathcal{H}^k(V) = \{0\}\)),

• Given two cochain complexes \(V,V^{\prime}\), a cochain map \(f =(f_k)\) (such as \(\diff^{\prime}_k f_k = f_{k+1} \diff_k\))

\[\begin{split}\begin{array}{ccccccccc} \cdots & \longrightarrow & V_{k-1} & \mbox{\shortstack{$\diff_{k-1}$ \\ $\longrightarrow$}} & V_{k} & \mbox{\shortstack{$\diff_k$ \\ $\longrightarrow$}} & V_{k+1} & \longrightarrow~\cdots \\ & & \downarrow f_{k-1} & & \downarrow f_{k} & & \downarrow f_{k+1} & & \\ \cdots & \longrightarrow & V_{k-1}^{\prime} & \mbox{\shortstack{$\diff_{k-1}^{\prime}$ \\ $\longrightarrow$}} & V_{k}^{\prime} & \mbox{\shortstack{$\diff_k^{\prime}$ \\ $\longrightarrow$}} & V_{k+1}^{\prime} & \longrightarrow~\cdots \end{array}\end{split}\]

• \(f_k\) maps \(k\)-cochains to \(k\)-cochains and \(k\)-coboundaries to \(k\)-coboundaries, thus induces a map \(\mathcal{H}^k(f) : \mathcal{H}^k(V) \longrightarrow \mathcal{H}^k(V^{\prime})\).

Let \(V^{\prime} \subset V\) be two cochain complexes,

• The inclusion \(\imath_V\) is a cochain map and thus induces a map of cohomology \(\mathcal{H}^k(V^{\prime}) \longrightarrow \mathcal{H}^k(V)\),

• If there exists a cochain projection of \(V\) onto \(V^{\prime}\), (this leads to \(\pi \circ \imath = \id_{V^{\prime}}\)) so \(\mathcal{H}^k(\pi) \circ \mathcal{H}^k(\imath) = \id_{\mathcal{H}^k(V^{\prime})}\).

  \[\begin{split}\begin{array}{ccccccc} \cdots & \longrightarrow & V_{k-1} & \mbox{\shortstack{$\diff_{k-1}$ \\ $\longrightarrow$}} & V_{k} & \longrightarrow~\cdots \\ & & \pi_{k-1} \downarrow \uparrow \imath & & \pi_{k} \downarrow \uparrow \imath & & \\ \cdots & \longrightarrow & V_{k-1}^{\prime} & \mbox{\shortstack{$\diff_{k-1}$ \\ $\longrightarrow$}} & V_{k}^{\prime} & \longrightarrow~\cdots \end{array}\end{split}\]

Thus, \(\mathcal{H}^k(\imath)\) is injective and \(\mathcal{H}^k(\pi)\) is surjective. Hence, if one of the cohomology spaces \(\mathcal{H}^k(V)\) vanishes, then so does \(\mathcal{H}^k(V^{\prime})\)

Cycles and boundaries of the De Rham complex

• \(k\)-cocycles

\[\mathfrak{Z}^k = \{ \omega \in H\Lambda^k(\Omega),~~ \diff \omega = 0 \}, ~~~ \mathfrak{Z}^{*k} = \{ \omega \in H^*\Lambda^k(\Omega),~~ \delta \omega = 0 \},\]

\[\mathfrak{Z}_0^k = \{ \omega \in H_0\Lambda^k(\Omega),~~ \diff \omega = 0 \}, ~~~ \mathfrak{Z}_0^{*k} = \{ \omega \in H_0^*\Lambda^k(\Omega),~~ \delta \omega = 0 \},\]

\[\]

• \(k\)-coboundaries

\[\mathfrak{B}^k = \diff H\Lambda^{k-1}(\Omega), ~~~ \mathfrak{B}^{* k} = \delta \Lambda^{k+1}(\Omega),\]

\[\mathfrak{B}_0^k = \diff H_0\Lambda^{k-1}(\Omega), ~~~ \mathfrak{B}_0^{* k} = \delta \Lambda_0^{k+1}(\Omega),\]

• each of the spaces of cycles is closed in \(\mathcal{H} \Lambda^k(\Omega)\) (\(\mathcal{H}^* \Lambda^k(\Omega)\)), as well in \(L^2 \Lambda^k(\Omega)\).

• each of the spaces of boundaries is closed in \(L^2 \Lambda^k(\Omega)\).

• let \(\perp\) denotes the orthogonal complement in \(L^2 \Lambda^k(\Omega)\),

\[\mathfrak{Z}^{k \perp} \subset \mathfrak{B}^{k \perp} = \mathfrak{Z}_0^{* k} , ~~~ \mathfrak{Z}^{* k \perp} \subset \mathfrak{B}^{* k \perp} = \mathfrak{Z}_0^{k}\]

\[\mathfrak{Z}_0^{k \perp} \subset \mathfrak{B}_0^{k \perp} = \mathfrak{Z}^{* k} , ~~~ \mathfrak{Z}_0^{* k \perp} \subset \mathfrak{B}_0^{* k \perp} = \mathfrak{Z}^{k}\]

The Hodge decomposition

There are two Hodge decompositions, with different boundary conditions,

1. \[L^2 \Lambda^k(\Omega) = \underbrace{\mathfrak{B}^{k}}_{\mathfrak{Z}_0^{* k\perp}} \oplus \underbrace{\mathfrak{H}^{k} \oplus \mathfrak{B}_0^{* k}}_{\mathfrak{Z}_0^{* k}=\mathfrak{B}^{k\perp}} = \overbrace{\mathfrak{B}^{k} \oplus \mathfrak{H}^{k}}^{\mathfrak{Z}^{k}=\mathfrak{B}_0^{* k\perp}} \oplus \overbrace{\mathfrak{B}_0^{* k}}^{\mathfrak{Z}^{k\perp}}\]
2. \[L^2 \Lambda^k(\Omega) = \underbrace{\mathfrak{B}_0^{k}}_{\mathfrak{Z}^{* k\perp}} \oplus \underbrace{\mathfrak{H}_0^{k} \oplus \mathfrak{B}^{* k}}_{\mathfrak{Z}^{* k}=\mathfrak{B}_0^{k\perp}} = \overbrace{\mathfrak{B}_0^{k} \oplus \mathfrak{H}_0^{k}}^{\mathfrak{Z}_0^{k}=\mathfrak{B}^{* k\perp}} \oplus \overbrace{\mathfrak{B}^{* k}}^{\mathfrak{Z}_0^{k\perp}}\]

Summary

DeRham sequence

here without boundary conditions

\[\mathbb{R} \hookrightarrow \Hgrad \xrightarrow{\quad \Grad \quad} \Hcurl \xrightarrow{\quad \Curl \quad} \Hdiv \xrightarrow{\quad \Div \quad} \Ltwo \xrightarrow{\quad} 0\]

Pullbacks

In the case where the physical domain \(\Omega := \mathcal{F}(\hat{\Omega})\) is the image of a logical domain \(\hat{\Omega}\) by a smooth mapping \(\mathcal{F}\) (at least \(\mathcal{C}^1\)), we have the following parallel diagrams

\[\begin{split}\begin{array}{ccccccc} \Hgrad & \xrightarrow{\quad \Grad \quad} & \Hcurl & \xrightarrow{\quad \Curl \quad} & \Hdiv & \xrightarrow{\quad \Div \quad} & \Ltwo \\ \igrad \Bigg\uparrow & & \icurl \Bigg\uparrow & & \idiv \Bigg\uparrow & & \iltwo \Bigg\uparrow \\ \HgradLogical & \xrightarrow{\quad \Grad \quad} & \HcurlLogical & \xrightarrow{\quad \Curl \quad} & \HdivLogical & \xrightarrow{\quad \Div \quad} & \LtwoLogical \\ % \end{array}\end{split}\]

Where the mappings \(\igrad, \icurl, \idiv\) and \(\iltwo\) are called pullbacks and are given by

\[\begin{split}\phi (x) :=& \igrad \hat{\phi} (\hat{x}) = \hat{\phi}(\mathcal{F}^{-1}(x)) \\ \Psi (x) :=& \icurl \hat{\Psi} (\hat{x}) = \left( D \mathcal{F} \right)^{-T} \hat{\Psi}(\mathcal{F}^{-1}(x)) \\ \Phi (x) :=& \idiv \hat{\Phi} (\hat{x}) = \frac{1}{J} D \mathcal{F} \hat{\Phi}(\mathcal{F}^{-1}(x)) \\ \rho (x) :=& \iltwo \hat{\rho} (\hat{x}) = \hat{\rho}(\mathcal{F}^{-1}(x))\end{split}\]

where \(D \mathcal{F}\) is the jacobian matrix of the mapping \(\mathcal{F}\).

Note

The pullbacks \(\igrad, \icurl, \idiv\) and \(\iltwo\) are isomorphisms between the corresponding spaces.

Operators correspondency

\[\begin{split}\begin{tabular}{|c||c|c|c|c|} \hline $\omega^k \in \Lambda^k(\Omega)$ & $k=0$ & $k=1$ & $k=2$ & $k=3$ \\ \hline $\diff \omega^k$ & $\Grad u$ & $\Curl \uu$ & $\Div \uu$ & $-$ \\ $\delta \omega^k$ & $-$ & $-\Div \uu$ & $\Curl \uu$ & $-\Grad u$ \\ $\mathfrak{i}_{\boldsymbol{\beta}} \omega^k$ & $-$ & $\boldsymbol{\beta} \cdot \uu$ & $\uu \times \boldsymbol{\beta}$ & $u \boldsymbol{\beta}$ \\ $\mathfrak{j}_{\boldsymbol{\beta}} \omega^k$ & $u \boldsymbol{\beta}$ & $-\uu \times \boldsymbol{\beta}$ & $\boldsymbol{\beta} \cdot \uu$ & $-$ \\ $L_{\boldsymbol{\beta}} \omega^k$ & $\boldsymbol{\beta} \cdot \Grad u$ & $\Grad \left(\boldsymbol{\beta} \cdot \uu \right) + \left(\Curl \uu \right) \times \boldsymbol{\beta}$ & $\Curl \left(\uu \times \boldsymbol{\beta} \right) + \boldsymbol{\beta} \Div \uu$ & $\Div \left( u \boldsymbol{\beta} \right)$ \\ $\mathcal{L}_{\boldsymbol{\beta}} \omega^k$ & $-\Div \left( u \boldsymbol{\beta} \right)$ & $-\Curl \left(\uu \times \boldsymbol{\beta} \right) - \boldsymbol{\beta} \Div \uu$ & $-\Grad \left(\boldsymbol{\beta} \cdot \uu \right) - \left(\Curl \uu \right) \times \boldsymbol{\beta}$ & $-\boldsymbol{\beta} \cdot \Grad u$ \\ \hline $\tr \omega^k$ & $u(\xx)$ & $\uu(\xx) \times \nn(\xx)$ & $\uu(\xx) \cdot \nn(\xx)$ & $-$ \\ \hline \hline $H \Lambda^k(\Omega)$ & $\Hgrad$ & $\Hcurl$ & $\Hdiv$ & $\Ltwo$ \\ $V_k$ & $\Vgrad$ & $\Vcurl$ & $\Vdiv$ & $\Vltwo$ \\ \hline \end{tabular}\end{split}\]

References