Exterior Calculus: Differential Forms

In elementary calculus, we learned $d{x}$ as an infinitesimal change, but now we are grown up😊so we would like to rephrase it as $1$-form. So, $x$ is refer to as a $0$-form. The higher order forms such as the $2$-form $d{A}$ and the $3$-form $d{V}$ are often thought of in-terms of their composite $1$-forms. This suggests, $1$-form is a building blocks of modern differential calculus which was pioneered by Élie Cartan.

General idea

  • Vector is a linear combination of basis vectors. For instance, $\vec{A} = A_{1} \hat{x} + A_{2}\hat{y} + A_{3}\hat{z}$.
  • 1-form is a linear combination of differentials. For instance, $A = A_{1} dx + A_{2} dy + A_{3} dz$.
  • Notice that, no arrow head on 1-form.
  • Use exterior products, to extend to higher dimensional forms (i.e. p-forms).
  • Completely antisymmetric $\begin{pmatrix} 0 \\ p \end{pmatrix}$ tensors, called p-forms.

Notation & conventions

  • $T_{ij\ldots k} = \tilde{T}(\vec{e}_{i}, \vec{e}_{j}, \ldots, \vec{e}_{k})$ where ${\vec{e}_{i}}$ is the set of basis vectors.
  • For $\begin{pmatrix} 0 \\ 2 \end{pmatrix}$ tensor, $T_{[ij]} = \frac{1}{2!} (T_{ij} - T_{ji})$.
  • Thus, $\begin{pmatrix} 0 \\ p \end{pmatrix}$ tensor, $T_{[i_{1}\ldots i_{p}]} = \frac{1}{p!}$(alternating sum over permutations of the indices $i_{1}$ to $i_{p}$).
    • For example, $T_{ijk} = \frac{1}{6} (T_{ijk} - T_{jik} + T_{jki} - T_{kji} + T_{kij} - T_{ikj})$
  • Tensor product of a $\tilde{T}$ of $\begin{pmatrix} 0 \\ p\end{pmatrix}$ tensor, and $\tilde{S}$ of $\begin{pmatrix} 0 \\ q\end{pmatrix}$ is $\tilde{T} \otimes \tilde{S}$ of $\begin{pmatrix} 0 \\ p + q \end{pmatrix}$ tensor.
  • $\tilde{T} \otimes \tilde{S}$ action on $(p+q)$ vector arguments is

    $\tilde{T} \otimes \tilde{S} (\vec{A}_{1}, \ldots, \vec{A}_{p}, \vec{B}_{1}, \ldots, \vec{B}_{q}) = \tilde{T}(\vec{A}_{1}, \ldots, \vec{A}_{p})\tilde{S}(\vec{B}_{1}, \ldots, \vec{B}_{q})$

Operators

  • Differential (exterior derivative): Takes $p$-forms as inputs and create $(p+1)$-forms; $d := \left( \frac{\partial }{\partial x}dx + \frac{\partial }{\partial y}dy + \frac{\partial }{\partial z}dz\right)\wedge$
  • $d^{n} \equiv d\circ d\circ d\ldots n~\text{times} := 0$
  • Exterior product (wedge product): $dx \wedge dy = -dy \wedge dx$
  • Hodge star operator: $\star : \Lambda^{p}(\Omega) \to \Lambda^{n-p}(\Omega)$ where $\Lambda^{p}(\Omega)$ is a space of diffferential p-forms on a smooth connected n-dimensional manifold $\Omega$.

    $$ \begin{align*} \star dx &= dy \wedge dz, \star\star dx = dx, dx = \star dy\wedge dz \\ \star dy &= dz \wedge dx \\ \star dz &= dx \wedge dy \end{align*} $$

  • co-differential (exterior anti-derivative): Takes $p$-forms as input and create $(p-1)$-forms; $\delta := \star d \star$
  • $\delta^{n} := 0$
  • Hodge Laplacian (Higher order Laplacian): Laplacian act on $p$-forms;

    $\Delta^{p} := (d + \delta)^{2} = d^{2} + d\delta + \delta d + \delta^{2} = d\delta + \delta d$

  • Musical operator (Musical Isomorphism):
    • Flat operator $\flat$: Transform vector fields into forms. i.e. $\vec{v} := \sum_{i = 1}^{n} f_{i} \frac{\partial}{\partial x_{i}} \to \vec{v}^{~\flat} \equiv v := \sum_{i=1}^{n}f_{i}dx_{i}$
    • Sharp operator $\sharp$: Transform forms into vector fields. i.e. $v := \sum_{i=1}^{n}f_{i}dx_{i} \to v^{\sharp} \equiv \vec{v} := \sum_{i = 1}^{n} f_{i} \frac{\partial}{\partial x_{i}}$

Operations

  • $d(U + V) = dU + dV$ where $U$ and $V$ are $p$-forms.
  • Leibniz product rule: $d(U \wedge V) = dU \wedge V + (-1)^{\text{deg(V)}} U \wedge dV$.
  • $d(f U) = d(f \wedge U) = df \wedge U + f \wedge dU$ where $f$ is a $0$-form.
  • Given $x$ is a $1$-form and $y$ is a vector field then, $(x^{\sharp})^{\flat} = x$ and $(y^{\flat})^{\sharp} = y$. These two operators cancel each other.
  • Applying $\star$ twice to a $p$-form $U$ will give back up to sign. i.e.

    $\star\star U = -1^{(n-p)p} U$ where $n$ is the dimension of the manifold.

  • Applying four times to $U$ always gives to identity. i.e.

    $\star\star\star\star U = U$

Algebraic topological jargons

NameMeaning
coboundary maps$A \in \mathbb{R}^{m\times n}, B\in \mathbb{R}^{n\times p}$
cochainselements of $\mathbb{R}$
cochains complex$\mathbb{R}^{p} \stackrel{B}{\to} \mathbb{R}^{n} \stackrel{A}{\to} \mathbb{R}^{m}$
cocycleselements of $\text{ker}(A)$
coboundarieselements of $\text{im}(B)$
cohomology classeselements of $\text{ker}(A)/\text{im}(B)$
harmonic cochainselements of $\text{ker}(A^* A + B B^*)$
Betti numbers$\text{dim}~\text{ker}(A^* A + B B^*)$
Hodge Laplacians$A^* A + B B^* \in \mathbb{R}^{n \times n}$
$x$ is closed$Ax = 0$
$x$ is exact$x = Bv$ for some $v \in \mathbb{R}^{p}$
$x$ is coclosed$B^{*}x = 0$
$x$ is coexact$x = A^* w$ for some $w \in \mathbb{R}^{m}$
$x$ is harmonic$(A^* A + B B*)x = 0$

Upshots

  • Poincare Lemma: $dV = 0 \Leftrightarrow V = dU$.
    • if $dV = 0$ then $V$ is said to be closed.
    • if $V = dU$ then $V$ is said to be exact.
  • Generalized Storke theorem: $\oint_{\partial \Omega} U = \partial_{\Omega} dU$.
  • Harmonic forms: if $p$-form $U$ is harmonic iff $\Delta U = 0$.
  • Helmholtz-Hodge decomposition theorem for p-form $\omega$: $\omega = d^{p-1} \phi + \delta^{p+1} \psi + h^{p}(\omega)$ where $\phi, \psi, $ and $h$ are scalar ($0$-form), $2$-form, and harmonic component ($p$-form), respectively.
  • Eigenvalue problem: $\Delta^{p} \omega = \lambda \omega$
  • Generalized forms: $-1$-form := a form of negative degree (Sparling 1997, Nurowski-Robinson 2001, 2002)

Applications

Classical vector calculus

  • Below is the exterior calculus equivalance to vector calculus.
    Classical differential operatorExterior differential operator
    $x$ is a scalar field (or just a function)$x$ is a $0$-form
    Gradient: $\nabla x$$(dx)^{\sharp}$
    Divergence: $\nabla \cdot \vec{A}$$\delta A$
    Curl: $\nabla \times \vec{A}$$(\star d A^{\flat})^{\sharp}$
    Scalar Laplacian: $\Delta x := \nabla\cdot\nabla x$$\Delta^{0} x := \delta d x$
    Vector (Helmholtz) Laplacian,
    $\Delta \vec{A} := \nabla (\nabla\cdot\vec{A}) - \nabla\times(\nabla\times\vec{A})$
    $\Delta^{2} A := (\delta A)^{\sharp} - (\star d((\star dA^{\flat})^{\sharp})^{\flat})^{\sharp} \equiv (d\delta + \delta d)A$
    Tensor Laplacian,
    $A_{\mu\nu;\lambda}^{\hspace{0.5cm};\lambda} = \frac{1}{\sqrt{|g_{\mu\nu}|}} (\sqrt{|g_{\mu\nu}|} g^{\mu\kappa} A_{\mu\nu,\kappa})_{,\mu}$
    $\Delta^{p} = d\delta + \delta d$

Electromagnetic fields

Classical differential operatorExterior differential operator
$\text{grad} f := \nabla f$$df$
$\text{curl} \vec{A} := \nabla \times \vec{A}$$dA$
$\text{div}\vec{B} := \nabla\cdot \vec{B}$$dB$
$\nabla\times \nabla f = 0$$d^{2} f = 0$
$\nabla\cdot\nabla\times\vec{A} = 0$$d^{2}A = 0$
Scalar Laplacian, $\Delta^{0} := \nabla\cdot\nabla \equiv \nabla^{2}$$\delta d f$
$\nabla\times\nabla\times \vec{A}$$\delta d A$

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Published on Mar 13, 2022

Last revised on Apr 3, 2022

References