Exterior Algebra **************** .. sectionauthor:: A. Ratnani Let :math:`V` be a real vector space of dimension :math:`n`. .. topic:: Definition, Alternating algebraic forms: For each :math:`k`, we define :math:`\Alt^k V` as the space of alternating :math:`k`-linear maps :math:`V \times \cdots \times V \rightarrow \mathbb{R}`. .. note:: * :math:`\Alt^0 = \mathbb{R}`, * :math:`\Alt^1 = V^{*}` is the dual space of :math:`V` (the space of covectors) .. topic:: Definition, Exterior product: For :math:`\omega \in \Alt^j` and :math:`\eta \in \Alt^k`, their exterior (wedge) product is given by: .. math:: (\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 :math:`v_i \in V`. Where the sum is over all permutations :math:`\sigma` of :math:`\{ 1,\cdots,j+k \}`, for which :math:`\sigma(1)< \cdots <\sigma(j)` and :math:`\sigma(j+1)< \cdots <\sigma(j+k)`. .. note:: * The exterior product is **bilinear**, **associative**, * **anti-commutative**: :math:`\eta \wedge \omega = (-1)^{jk} \omega \wedge \eta` for all :math:`\omega \in \Alt^j` and :math:`\eta \in \Alt^k`. .. topic:: Definition, Grassmann Algebra: Grassmann Algebra is defined by: .. math:: \Alt V := \bigoplus_k \Alt^k V This is a **anti-commutative graded algebra**. Also called **Exterior Algebra** of :math:`V^{*}` In the case of :math:`V=\mathbb{R}^n`, we have: * :math:`\Alt V^0 \sim \mathbb{R}`, * :math:`\Alt V^1 \sim \mathbb{R}^n`, * :math:`\Alt V^{n-1} \sim \mathbb{R}^n`, using Riesz representation theorem, * :math:`\Alt V^n \sim \mathbb{R}`, using the map :math:`v \longmapsto \det(v,v_1,\cdots,v_{n-1})`. Basis ^^^^^ Let :math:`v_1,\cdots,v_n` be a basis of :math:`V` and :math:`\mu_1,\cdots,\mu_n` the associated dual basis for :math:`V^*` (:math:`\mu_i(v_j) = \delta_{ij}`). For any increasing permutations :math:`\sigma, \rho : \{ 1,\cdots,k \} \longrightarrow \{ 1,\cdots,n \}`, we have: .. math:: \mu_{\sigma(1)} \wedge \cdots \wedge \mu_{\sigma(k)} (v_{\rho(1)}, \cdots, v_{\rho(k)}) = \chi_{\sigma,\rho} thus the :math:`\binom {n}{k}` algebraic :math:`k`-forms :math:`\mu_{\sigma(1)} \wedge \cdots \wedge \mu_{\sigma(k)}`, form a basis for :math:`\Alt^k V` and :math:`\dim \Alt^k V = \binom {n}{k}`. .. topic:: Definition, Interior product: Let :math:`\omega` be a :math:`k`-form, and :math:`v \in V`. The **interior product** of :math:`\omega` and :math:`v` is the :math:`(k-1)`-form :math:`\omega \lrcorner v` defined by: .. math:: \omega \lrcorner v (v_1,\cdots,v_{k-1}) = \omega (v,v_1,\cdots,v_{k-1}) * We have for :math:`\omega \in \Alt^k V`, :math:`\eta \in \Alt^l V` and :math:`v \in V`: .. math:: (\omega \wedge \eta) \lrcorner v = (\omega \lrcorner v)\wedge \eta + (-1)^k \omega \wedge (\eta \lrcorner v) .. topic:: Definition, Inner product: If :math:`V` is has an inner product, then :math:`\Alt^k V` is endowed with an inner product given by: .. math:: (\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 :math:`\rho : \{ 1,\cdots,k \} \longrightarrow \{ 1,\cdots,n \}`, and :math:`v_1, \cdots,v_n` is any orthonormal basis. Orientation and Volume form ^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. todo:: add Orientation and Volume form .. topic:: Definition, Pullback: A linear transformation of vector spaces :math:`L: V \rightarrow W` induces a transformation :math:`L^{*}: \Alt W \rightarrow \Alt V`, called the **pullback**, and given by: .. math:: 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 :math:`U \xrightarrow{~K~} V \xrightarrow{~L~} W` then, .. math:: \Alt W \xrightarrow{~K^{*}~} \Alt V \xrightarrow{~L^{*}~} \Alt U * :math:`L^{*} (\omega \wedge \eta) = L^{*} \omega \wedge L^{*} \eta` Let V be a subspace of W. For the inclusion :math:`\imath_V : V \longrightarrow W`, we can define its pullback :math:`\imath_V^{*}`: this is a **surjection** of :math:`\Alt W` onto :math:`\Alt V`. If W has an inner product and :math:`\pi_V : W \longrightarrow V` is the orthogonal projection. We can define its pullback :math:`\pi_V^{*}` : this an **injection** of :math:`\Alt V` onto :math:`\Alt W`. Let us consider the composition : :math:`W` \shortstack{:math:`\pi_V` \\ :math:`\longrightarrow`} :math:`V` \shortstack{:math:`\imath_V` \\ :math:`\longrightarrow`} :math:`W`, and its pullback :math:`\pi_V^* \imath_V^*`. .. topic:: Definition, The tangential and normal parts: * :math:`\pi_V^* \imath_V^*` associates for each :math:`\omega \in \Alt^k` its **tangential** part :math:`\omega_{\parallel}` with respect to :math:`V` : .. math:: (\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. * :math:`\omega - \pi_V^* \imath_V^* \omega` associates for each :math:`\omega \in \Alt^k` its **normal** part :math:`\omega_{\perp}` with respect to :math:`V`. The **tangential part** of :math:`\omega` vanishes if and only if the image of :math:`\omega` in :math:`\Alt^k V` vanishes. Let :math:`V` be an oriented inner product space, with volume form :math:`\mbox{vol}`. Let :math:`\omega \in \Alt^k V`. We can define a new linear map :math:`L_{\omega}` as the composition of :math:`\Alt^{n-k} V \longrightarrow \Alt^n V` such as: .. math:: \mu \longmapsto \omega \wedge \mu and the canonical isomorphism of :math:`\Alt^n V` onto :math:`\mathbb{R}`, and using the Riesz representation theorem, there exists an element :math:`\star \omega \in \Alt^{n-k} V` such that : :math:`L_{\omega} (\mu) = (\star \omega , \mu)`, *i.e.*: .. math:: \omega \wedge \mu = (\star \omega , \mu) \mbox{vol}, ~~~\omega \in \Alt^{k}, ~\mu \in \Alt^{n-k} .. topic:: Definition, The Hodge star operation: The linear map which maps :math:`\Alt^k V` onto :math:`\Alt^{n-k} V` :math:`\omega \longmapsto \star \omega` is called the **Hodge star** operator. * If :math:`e_1,\cdots,e_n` is any positively oriented orthonormal basis, and :math:`\sigma` a permutation, we have .. math:: \omega(e_{\sigma(1)}, \cdots, e_{\sigma(k)}) = (\mathrm{sign} \sigma) \star \omega(e_{\sigma(k+1)}, \cdots, e_{\sigma(n)}) * :math:`\star \star \omega = (-1)^{k(n-k)} \omega, ~~~\forall \omega \in \Alt^k V`, thus the Hodge star is an **isometry**. * :math:`(\star \omega)_{\parallel} = \star (\omega_{\perp})` and :math:`(\star \omega)_{\perp} = \star (\omega_{\parallel})` * the image of :math:`\star \omega` in :math:`\Alt^k V` vanishes if and only if :math:`\omega_{\perp}` vanishes. .. math:: \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} .. \caption{Correspondence} .. math:: \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} .. \caption{Exterior product} .. math:: \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} .. \caption{Pullback by a linear map L :$\mathbb{R}^3 \longrightarrow \mathbb{R}^3$} .. math:: \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} .. \caption{Interior product with a vector $v \in \mathbb{R}^3$} .. math:: \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} .. \caption{Inner product and volume form} .. math:: \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} .. \caption{Hodge star} Exterior Calculus on manifolds and Differential forms ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Let :math:`\Omega` be a smooth manifold, of dimension :math:`n`. * :math:`\forall x \in \Omega` we denote by :math:`T_x \Omega` the tangent space. This is a vector space of dimension :math:`n`, * tangent bundle :math:`\{ (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 :math:`(x,\mu)` with :math:`x \in \Omega` and :math:`\mu \in \Alt^k T_x \Omega`. * a **differential** :math:`k`-form :math:`\omega` is a section of this bundle. This is a map which associates to each :math:`x \in \Omega` an element :math:`\omega_x \in \Alt^k T_x \Omega`, * if the map :math:`\mathcal{L}_{\omega}^k : x \longmapsto \omega_x (v_1(x), \cdots, v_k(x))` is smooth (whenever :math:`v_i` are smooth), we say that :math:`\omega` is a smooth differential :math:`k`-form, * we define :math:`\Lambda^k(\Omega)` the space of all smooth :math:`k`-forms on :math:`\Omega`, * :math:`\Lambda^0(\Omega) = \mathcal{C}^{\infty}(\Omega)`, * if the map :math:`\mathcal{L}_{\omega}^k` is :math:`\mathcal{C}^{m}(\Omega)`, we define differential :math:`k`-forms with less smoothness :math:`\mathcal{C}^{m} \Lambda^k (\Omega)`. Let :math:`\Omega` be a smooth manifold, of dimension :math:`n`. .. topic:: Exterior product: if :math:`\omega \in \Lambda^k(\Omega)` and :math:`\eta \in \Lambda^j(\Omega)`, we may define :math:`\omega \wedge \eta` as :math:`(\omega \wedge \eta)_x = \omega_x \wedge \eta_x` and the Grassmann algebra :math:`\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. .. topic:: Exterior differentiation: For each :math:`\omega \in \Lambda^k(\Omega)`, can define the :math:`(k+1)`-form :math:`\diff \omega \in \Lambda^{k+1}(\Omega)`, such as: .. math:: \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 :math:`+1`, of :math:`\Lambda(\Omega)` onto :math:`\Lambda(\Omega)`. We have the following properties: * :math:`\diff \circ \diff = 0` * :math:`\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 :math:`\phi` be a smooth map of :math:`\Omega` onto :math:`\Omega^{\prime}`. Then :math:`\phi^*(\omega \wedge \eta) = \phi^*(\omega) \wedge \phi^*(\eta)` and :math:`\phi^* (\diff \omega) = \diff (\phi^* \omega)`, * (Interior product) the interior product of a differential :math:`k`-form :math:`\omega` with a vector field :math:`v`, * we obtain a :math:`(k-1)`-form by : :math:`(\omega \lrcorner v)_x := \omega_x \lrcorner v_x`, * (Trace operator) the pullback :math:`i_{\partial \Omega}^*` of :math:`i_{\partial \Omega}` is the trace operator :math:`\trace` .. topic:: Integration: * If :math:`f` is an oriented, piecewise smooth :math:`k`-dimensional submanifold of :math:`\Omega`, and :math:`\omega` is a continuous :math:`k`-form, then th integral :math:`\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 :math:`L^2`-inner product of two differential :math:`k`-forms on an oriented Riemannian manifold :math:`\Omega` is defined as : .. math:: (\omega,\eta)_{L^2 \Lambda^k} = \int_{\Omega} (\omega_x,\eta_x) \volume = \int \omega \wedge \star \eta The completion of :math:`\Lambda^k(\Omega)` in the corresponding norm defines the Hilbert space :math:`L^2 \Lambda^k(\Omega)`. We have the following results: * (Integration) if :math:`\phi` is an orientation-preserving diffeomorphism, then .. math:: \int_{\Omega} \phi^* \omega = \int_{\Omega^{\prime}} \omega, ~~~ \forall \omega \in \Lambda^n(\Omega^{\prime}) .. topic:: Theorem, Stokes theorem: If :math:`\Omega` is an oriented :math:`n`-manifold with boundary :math:`\partial \Omega`, then .. math:: \int_{\Omega} \diff \omega = \int_{\partial \Omega} \trace \omega, ~~~ \forall \omega \in \Lambda^{n-1}(\Omega) .. topic:: Theorem, Integration by parts: If :math:`\Omega` is an oriented :math:`n`-manifold with boundary :math:`\partial \Omega`, then .. math:: \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: * :math:`H^s \Lambda^k(\Omega)` is the space of differential :math:`k`-forms such that :math:`\mathcal{L}_{\omega}^k \in H^s(\Omega)`. * :math:`H \Lambda^k(\Omega) = \{ \omega \in L^2 \Lambda^k(\Omega),~~ \diff \omega \in L^2 \Lambda^{k+1}(\Omega) \}`. The associated norm is : .. math:: \| \omega \|_{H \Lambda^k}^2 = \| \omega \|_{H \Lambda}^2 := \| \omega \|_{L^2 \Lambda^k}^2 + \| \diff \omega \|_{L^2 \Lambda^{k+1}}^2 * :math:`H \Lambda^{0}(\Omega)` coincides with :math:`H^1 \Lambda^{0}(\Omega)`, * :math:`H \Lambda^{n}(\Omega)` coincides with :math:`L^2 \Lambda^{n}(\Omega)`, * for :math:`0 < k < n`, we have :math:`H^1 \Lambda^k(\Omega) \subset H \Lambda^k(\Omega) \subset L^2 \Lambda^k(\Omega)`, strictly. .. math:: \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} .. \caption{Correspondences between differential forms in $3$D, and scalar/vector fields.} Cohomology and De Rham Complex ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The De Rham complex is the sequence of spaces and mappings .. math:: 0 \xrightarrow{\quad} \Lambda^0(\Omega) \xrightarrow{~\diff~} \Lambda^1(\Omega) \xrightarrow{~\diff~} \cdots \xrightarrow{~\diff~} \Lambda^n(\Omega) \xrightarrow{\quad} 0 Since, :math:`\diff \circ \diff = 0`, we have .. math:: \mathcal{R}(\diff : \Lambda^{k-1}(\Omega) \longrightarrow \Lambda^k(\Omega)) \subset \mathcal{N}(\diff : \Lambda^{k}(\Omega) \longrightarrow \Lambda^{k+1}(\Omega)) If :math:`\Omega` is an oriented Riemannian manifold, we have the following cohomology: .. math:: 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* :math:`\delta : \Lambda^{k}(\Omega) \longrightarrow \Lambda^{k-1}(\Omega)` is defined as: .. math:: \star \delta \omega = (-1)^k \diff \star \omega,~~~ \omega \in \Lambda^k(\Omega) * we have .. math:: (\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), * :math:`\delta` is a graded linear operator of degree :math:`-1`. * :math:`\delta` is the formal adjoint of :math:`\diff` whenever :math:`\omega` or :math:`\eta` vanishes near the boundary. * we define the spaces .. math:: H^* \Lambda^k(\Omega) = \{ \omega \in L^2 \Lambda^k(\Omega),~~ \delta \omega \in L^2 \Lambda^{k-1}(\Omega) \}. we have :math:`H^* \Lambda^k(\Omega) = \star H \Lambda^{n-k}(\Omega)`. * we obtain the dual complex .. math:: 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 :math:`\Lambda_0^k(\Omega)` be the subspace of :math:`\Lambda^k(\Omega)` of smooth :math:`k`-forms with compact support. We have :math:`\diff \Lambda_0^k \subset \Lambda_0^{k+1}`. The De Rham complex with the compact support is .. math:: 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 :math:`\Lambda_0^k(\Omega)` in :math:`H \Lambda^k(\Omega)` is .. math:: H_0 \Lambda^k(\Omega) = \{ \omega \in H \Lambda^k(\Omega),~~ \trace \omega =0\}. The :math:`L^2` version of the last complex is .. math:: 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 .. topic:: Definition, Harmonic forms: The harmonic :math:`k`-forms are the differential :math:`k`-forms that verify the differential equations .. math:: \left\{ \begin{aligned} \diff \omega &=& 0,\\ \delta \omega &=& 0,\\ \trace \star \omega &=& 0.\\ \end{aligned} \right. this defines the following space, .. math:: \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, .. math:: \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, :math:`\star \mathfrak{H}^k (\Omega) = \mathfrak{H}_0^{n-k} (\Omega)`. .. topic:: Proposition, Poincaré duality: There is an isomorphism between the :math:`k` th De Rham cohomology space and the :math:`(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 .. TODO .. .. math:: .. .. \cdots \longrightarrow V_{k-1}~\mbox{\shortstack{:math:`\diff_{k-1}` \\ :math:`\longrightarrow`}}~V_{k} \mbox{\shortstack{:math:`\diff_k` \\ :math:`\longrightarrow`}}~V_{k+1}~\longrightarrow~\cdots,~~~~\mbox{with}~ \diff_{k+1} \circ \diff_k = 0. .. * :math:`k`-cocycles :math:`\mathfrak{Z}^k := \mathcal{N}(d_k)`, * :math:`k`-coboundaries :math:`\mathfrak{B}^k := \mathcal{R}(d_{k-1})`, * :math:`k`-cohomology :math:`\mathcal{H}^k(V) := \mathfrak{Z}^k / \mathfrak{B}^k`, * we say that the sequence is **exact**, if the **cohomology vanishes** (*i.e.* :math:`\forall~k,~~ \mathcal{H}^k(V) = \{0\}`), * Given two cochain complexes :math:`V,V^{\prime}`, a **cochain map** :math:`f =(f_k)` (such as :math:`\diff^{\prime}_k f_k = f_{k+1} \diff_k`) .. math:: \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} * :math:`f_k` maps :math:`k`-cochains to :math:`k`-cochains and :math:`k`-coboundaries to :math:`k`-coboundaries, thus induces a map :math:`\mathcal{H}^k(f) : \mathcal{H}^k(V) \longrightarrow \mathcal{H}^k(V^{\prime})`. Let :math:`V^{\prime} \subset V` be two cochain complexes, * The inclusion :math:`\imath_V` is a cochain map and thus induces a map of cohomology :math:`\mathcal{H}^k(V^{\prime}) \longrightarrow \mathcal{H}^k(V)`, * If there exists a cochain projection of :math:`V` onto :math:`V^{\prime}`, (this leads to :math:`\pi \circ \imath = \id_{V^{\prime}}`) so :math:`\mathcal{H}^k(\pi) \circ \mathcal{H}^k(\imath) = \id_{\mathcal{H}^k(V^{\prime})}`. .. math:: \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} Thus, :math:`\mathcal{H}^k(\imath)` is **injective** and :math:`\mathcal{H}^k(\pi)` is **surjective**. Hence, if one of the cohomology spaces :math:`\mathcal{H}^k(V)` vanishes, then so does :math:`\mathcal{H}^k(V^{\prime})` Cycles and boundaries of the De Rham complex ____________________________________________ * :math:`k`-cocycles .. math:: \mathfrak{Z}^k = \{ \omega \in H\Lambda^k(\Omega),~~ \diff \omega = 0 \}, ~~~ \mathfrak{Z}^{*k} = \{ \omega \in H^*\Lambda^k(\Omega),~~ \delta \omega = 0 \}, .. math:: \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 \}, .. math:: * :math:`k`-coboundaries .. math:: \mathfrak{B}^k = \diff H\Lambda^{k-1}(\Omega), ~~~ \mathfrak{B}^{* k} = \delta \Lambda^{k+1}(\Omega), .. math:: \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 :math:`\mathcal{H} \Lambda^k(\Omega)` (:math:`\mathcal{H}^* \Lambda^k(\Omega)`), as well in :math:`L^2 \Lambda^k(\Omega)`. * each of the spaces of boundaries is closed in :math:`L^2 \Lambda^k(\Omega)`. * let :math:`\perp` denotes the orthogonal complement in :math:`L^2 \Lambda^k(\Omega)`, .. math:: \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} .. math:: \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. .. math:: 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. .. math:: 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 ^^^^^^^ .. math:: \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} .. \caption{Correspondences between differential forms in $3$D, and scalar/vector fields.} .. rubric:: References .. bibliography:: refs_feec.bib :cited: