Exterior Calculus

The theory of exterior (differential) calculus can be seen as an algebraic organization of differential geometry, initially proposed by Élie Cartan at the turn of the XXth century.

Differential forms

The theory of exterior calculus is built around the concept of (differential) k-form; which corresponds to a skew-symmetric, fully covariant tensor field of rank k, defined within the cotangent bundle of a n-dimensional manifold ($n\geq k$), i.e.,: $$\omega = \omega_{i\dots k}\boldsymbol{dx}^i\otimes\dots\otimes\boldsymbol{dx}^k \quad \text{s.t.} \quad \omega_{p(i\dots k)} = (-1)^{\sigma(p)} \omega_{i\dots k},$$ where $p$ corresponds to any permutation of the k indices and $\sigma(p) = \pm 1$ to its parity.

In this framework, linear forms — aka covectors — correspond to the specific case of 1-forms. Exterior calculus can therefore be considered as a generalization of vector calculus in arbitrary dimensions.

Moreover, as covectors corresponds to linear forms acting on vectors in the 1D case; k-forms extend this interpretation to the k-dimensional case. They correspond to k-linear operators assigning scalar values to k-vectors: $$\begin{array}{crl} \omega^{(k)} \colon & \mathcal{T}^k_{\mathcal{M}} &\to \mathbb{R} \\ & \boldsymbol{v}^{(k)} &\mapsto\omega^{(k)}\left(\boldsymbol{v}^{(k)}\right), \end{array}$$ where $\boldsymbol{v}^{(k)}$ stands for a k-vector (1) defined in the kD extended tangent bundle

1. This notion will be explicited in a section below.

$$\mathcal{T}^k_{\mathcal{M}} = \underbrace{\mathcal{T}_{\mathcal{M}}\otimes\dots\otimes\mathcal{T}_{\mathcal{M}}}_{\times k},$$ of the considered manifold $\mathcal{M}$.

Given a nD manifold $\mathcal{M}$, k-forms defined on it, k≤n, form a vector space, noted $\Lambda^k(\mathcal{M})$ hereafter.

In particular, this allows to extends classic operators of "vectorial" differential geometry, such as gradient, divergence and curl to higher dimensions. This extension is made possible by the definition of a dedicated derivation operator: The exterior derivative, see below.

A graded algebraic structure

The generalization of vector calculus to higher dimensions is based on the algebraic structure of k-forms. Indeed, (k+l)-forms can be built from k-dimensional and l-dimensional ones using the wedge product and noted $\wedge$: $$\omega^{(k+l)} = \alpha^{(k)}\wedge\beta^{(l)}.$$ Equipped with this wedge product, all the k-forms defined on a n-dimensional manifold consitute a graded algebra, called the exterior algebra $$\Lambda(\mathcal{M})=\Lambda^0(\mathcal{M})\oplus\Lambda^1(\mathcal{M})\oplus \cdots\oplus\Lambda^n(\mathcal{M}).$$ This graded algebra is of finite dimension, for the degree (k) of a k-form cannot exceed the dimension of the manifold it is defined upon, $k\leq n$.

Within this graded algebra, the set of k-forms defined on a nD manifold (for a given value of 0≤k≤n), corresponds to a $\binom{n}{k}$-dimensional vector space, where k-forms can be expanded over the canonical base forms: $$\omega^{(k)} = \omega_{i_1\dots i_k}\boldsymbol{dx}^{i_1}\wedge\dots\wedge\boldsymbol{dx}^{i_k},$$ where the Einstein summation convention has been used for concision and runs over the permutations of k indices ($i_1\dots i_k$) over n.

Exterior derivative

The exterior derivative is a linear operator that maps k-forms to (k+1)-forms. It is noted $d$ and extends the notion of differentiation to differential forms of arbitrary degrees.

This operator can be defined by a series of axioms it must follow:

Match the usual derivative on 0-formLinearityLiebniz rule2-nilpotency

For a 0-form , i.e. a smooth function $f\colon \mathcal{M}\to \mathbb{R}$, it corresponds to the usual differential: $$df = \frac{\partial f}{\partial x^i}dx^i$$

It must be linear, i.e. given a k-form $\alpha$ and an l-form $\beta$: $$d(a\alpha + b\beta) = ad\alpha + bd\beta \quad\forall a,b\in\mathbb{R}.$$

As differential forms as skew-symmetric, it must follow the Liebniz rule: $$d(\alpha\wedge\beta) = d\alpha\wedge\beta + (-1)^k\alpha\wedge d\beta \quad\forall (\alpha,\beta)\in\Lambda^k(\mathcal{M})\times\Lambda^l(\mathcal{M}).$$

Finally it must be 2-nilpotent: $$d(d(\alpha)) = 0,\quad \forall \alpha\in\Lambda^k(\mathcal{M}).$$

Within a coordinate system defined on the considered manifold, it can be expressed as: $$\begin{array}{crl} d\colon & \Lambda^k &\to \Lambda^{k+1} \\ & \omega &\mapsto d\omega \end{array} \quad\text{s.t.}\quad d\omega = \underset{I_k,i}\sum\frac{\partial \omega_{I_k}}{\partial x^i} dx^{I_k}\wedge dx^i,$$

where $I_k$ corresponds to a set of indices corresponding to k variables taken amongst the n ones needed to parametrize $\mathcal{M}$. We sum over all the possible sets, i.e. $\binomial{k,n}$, and $i$ is an index runnning over all n variables.

Stokes-Cartan theorem

The cornerstone of the theory is the Stokes-Cartan Theorem, which binds together the three main ingredients of exterior calculus: The integral of a k-form, $\omega^{(k)}$, over the boundary, $\partial\sigma^{(k+1)}$, of some (k+1)-dimensional oriented sub-manifold, $\sigma^{(k+1)}$, equals the integral of its exterior derivative, $d\omega^{(k)}$, over the whole sub-manifold: $$\int_{\sigma^{(k+1)}}d\omega^{(k)} = \int_{\partial\sigma^{(k+1)}}\omega^{(k)}.$$ This is a very powerful results for it provides a unifying view encompassing some major theorems in differential calculus and geometry, e.g. the second fundamental theorem of calculus, the Kelvin-Stokes theorem, Ostrogradsky's theorem, Green's theorem.

• Algebraic interpretation Given a nD oriented manifold, k-forms defined on it constitute a vector space, dual to the one formed by k-vectors. Consequently the Riesz representation theorem ensures that the integration of k-forms over k-vectors defines an inner product, often refered to as their "natural pairing": $$\begin{array}{c} \forall \sigma,\omega \in \mathcal{T}^{k}(\mathcal{M})\times\Lambda^{k}(\mathcal{M}),\; \\ \\ \omega(\sigma)=\langle \sigma, \omega\rangle = \int_\sigma\omega, \end{array}$$ for it is obviously linear in both $\sigma$ & $\omega$.

  Considering a (k+1)-vector $\sigma$ and a k-form $\omega$, we can rewrite the Stokes-Cartan theorem as: $$\langle \sigma, d\omega\rangle = \langle \partial\sigma, \omega\rangle % \quad\Longleftrightarrow\quad % d\omega(\sigma) = \omega(\partial\sigma).$$ This algebraic interpretation is another pivotal concept in exterior calculus and DEC for it states that the exterior derivative is the adjoint operator of the boundary operator.

Hodge duality

The finite dimension of the graded algebra of k-forms defined on a nD manifold $\mathcal{M}$ leads to another key concept in exterior calculus: Hodge duality. Given a k-form, $\alpha \in \Lambda^k(\mathcal{M})$, acting on the k-dimensional subset $\Omega\subset\mathcal{M}$; we can construct a dual (n-k)-form, noted $*\alpha$ here after, defined on $\star\Omega$ (1), verifying:

1. The orthogonal complement of the initial k-dim subset.

$$\int_{\star\Omega}*\alpha = \int_{\Omega}\alpha. \tag{1}$$ The isomorphism that maps k-forms to (n-k)-ones is called the Hodge star, and is noted $*(\cdot)$: $$\begin{array}{crl} *\colon & \Lambda^{k}(\mathcal{M}) \to & \Lambda^{(n-k)}(\mathcal{M}) \\ & \alpha \mapsto & *\alpha. \end{array}$$

Example

One ubiquituous application of this concept is the definition of the vectorial product in the usual 3D euclidean space. Indeed, when we write $$\boldsymbol{a}\wedge\boldsymbol{b} = \boldsymbol{c},$$ we implicitly:

• Assimilate vectors and covectors, which is fine in euclidean spaces.
• Associate to a 2-form, the lhs in the above formulae, defined within the plane spanned by the two (co)vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ to a 1-form, the rhs, spanning the 1D space orthogonal to the forementioned plane.

Practically, if we consider, still in 3D, the basic forms: $dx, dy, dz$, we have: $$\begin{array}{cc} *dx = dy\wedge dz,& *\left(dx\wedge dy\right) = dz \\ *dy = dz\wedge dx,& *\left(dy\wedge dz\right) = dx \\ *dz = dx\wedge dy,& *\left(dz\wedge dx\right) = dy \\ *\left(dx\wedge dy\wedge dz\right) = 1 \end{array}$$

One noteworthy property of the Hodge operator is that applied twice to a k-form $\omega$ defined on a nD manifold $\mathcal{M}$, it yields back the k-form up to a sign: $$\forall\omega\in\Lambda^k(\mathcal{M}),\: **\omega = (-1)^{k(n-k)}\omega.$$

Musical isomorphisms

As mentionned above, k-forms and k-vectors defined on a nD manifold $\mathcal{M}$ constitute dual vector spaces. The musical isomorphisms, "sharp" and "flat" (noted respectively $\cdot^{\sharp}, \cdot^{\flat}$), enable to pass from one to the other: $$\mathbb{V}^k\left(\mathcal{M}\right) \underset{\sharp}{\overset{\flat}{\rightleftarrows}} \Lambda^k\left(\mathcal{M}\right)$$

In the usual index-based formalization of differential geometry, given a manifold $\left(\mathcal{M},\boldsymbol{g}\right)$, they simply correspond to a contraction with the metric tensor $\boldsymbol{g}$ (1):

1. The origin of the poetic name "musical" isomorphisms is apparent when looking at the formulae below: As sharp and flat respectively raise and lower tones in music their algebraic counterparts raise and lower indices, transforming covectors (aka 1-forms) into vectors and vice-versa.

$$\begin{cases} \forall \boldsymbol{v}\in\mathbb{V}\left(\mathcal{M}\right), \boldsymbol{v}=v^i\boldsymbol{e}_i &\Rightarrow \boldsymbol{v}^{\flat} = v_i\boldsymbol{e}^i,\; \text{with}\; v_i = g_{ij}v^j,\\ \forall \boldsymbol{\omega}\in\Lambda\left(\mathcal{M}\right), \boldsymbol{\omega}=\omega_i\boldsymbol{e}^i &\Rightarrow \boldsymbol{\omega}^{\sharp} = \omega^i\boldsymbol{e}_i,\; \text{with}\; \omega^i = g^{ij}\omega_j,\\ \end{cases}$$ where $g_{ij}, g^{ij}$ (resp.) are the metric coefficients in there twice covariant (contravariant) version.

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In order to truly harnest this powerful tool in numerical simulations, a discrete version of this theory is needed. This is what the next section is about.