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 referred 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 block 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 arrowhead 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

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 operator	Exterior 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