Differential Forms

Remark (Motivation via Integration)

Differential $k$-forms are fundamentally objects designed to be integrated over $k$-dimensional oriented surfaces. The standard integrals from multivariable calculus are actually integrations of differential forms:

1. Line Integrals (1-forms): Integrating a vector field $\mathbf{v}$ along a curve $\Gamma$ gives: $${\int }_{\Gamma }\mathbf{v}\cdot d\mathbf{l}={\int }_{\Gamma }\underset{\text{differential 1-form}}{\underbrace{v_{1}dx+v_{2}dy+v_{3}dz}}$$ The integrand is a differential 1-form, which corresponds to a covector field.
2. Flux Integrals (2-forms): Integrating a vector field $\mathbf{w}$ across a surface $S$ gives: $${\int }_S\mathbf{w}\cdot \mathbf{n}dA={\int }_S\underset{\text{differential 2-form}}{\underbrace{w_{1}dydz+w_{2}dzdx+w_{3}dxdy}}$$ The area element here is skew-symmetric (i.e., $dxdy=-dydx$), which motivates the definition of 2-forms as skew-symmetric tensors in $T^{\ast }M\wedge T^{\ast }M$.
3. Volume Integrals (3-forms): Integrating a scalar function $f$ over a volume $V$ gives: $${\int }_{V}fdxdydz$$ where $dxdydz=-dydxdz$.

This integration-based intuition naturally leads to the formal definition of differential forms as skew-symmetric tensors on a manifold.

Definition (Differential Form)

Let $M$ be a Manifold and $V=T_{p}M$ be its tangent space at a point $p$. A differential $k$-form is a skew-symmetric tensor product of covectors.

Mathematically, it resides in the exterior power of the dual space (a subspace of the tensor product space):

$$\omega \in \underset{k}{\underbrace{V^{\ast }\wedge \cdots \wedge V^{\ast }}}=\stackrel{k}{\bigwedge }{V}^{\ast }\subset \stackrel{k}{⨂}{V}^{\ast }$$

If the basis for the dual space $V^{\ast }$ consists of the differentials of coordinate functions $d{x}^1,\dots ,d{x}^n$, then the standard basis for ${\bigwedge }^k{V}^{\ast }$ is given by the wedge products:

$$d{x}^I=d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_k}$$

where $i_1<\cdots <i_k$. And $I=(i_1,\dots ,i_k)$.

(For the algebraic properties of differential forms, including their dimensions, wedge products, pairings, and interior products, see Exterior Algebra).

Definition (Integration of k-form Field)

Let $\omega \in \Gamma ({\bigwedge }^k{T}^{\ast }M)={\Omega }^k(M)$ be a $k$-form field, and let $S$ be a $k$-dimensional surface. The integral of $\omega$ over $S$ is defined via a Riemann sum of the pairing over infinitesimal elements:

$${\int }_S\omega =\underset{\text{partition refines}}{\lim }\sum _{\text{all elements}}\omega {|}_p[[v_1,\dots ,v_k]]$$

where $v_1,\dots ,v_k$ span the tangent elements of the partition at point $p$.

Unpacking the Notation

• $T^{\ast }M$ is the cotangent bundle (the collection of all covector spaces at every point on the Manifold).
• ${\bigwedge }^k{T}^{\ast }M$ is the bundle containing all possible $k$-forms at every point.
• $\Gamma (\dots )$ denotes the space of “sections” of a bundle. Taking a section means smoothly assigning one specific $k$-form to each point on the manifold.
• ${\Omega }^k(M)$ is simply the standard shorthand notation for this space of sections. Therefore, $\omega \in {\Omega }^k(M)$ is the rigorous way to state that $\omega$ is a continuous $k$-form field.

Intuition

Imagine that $k$ is actually $3$, so that we are in a 3D world. In particular, consider the Earth’s surface, and that $M$, our Manifold, is a zoomed in part of the surface that looks flat (looks like a Euclidean Space at each point, in 2D). We want to integrate a 2-form over a patch of grass or water, say $S$.

At any point in the $p$ in the manifold, there is a tangent space $T_{p}M$. A vector field (a section of $TM$) assigns a physical direction and magnitude to each every point (like wind direction). We get two vectors $v_1,v_2$ which span the tangent elements of the partition at $p$.

If these two vectors span the grid on $S$, what is the $k-$form field $\omega \in {\Omega }^3(M)$? We can think of the $k-$form field $\omega$ as a continous field of wind sensors across the surface. At every point $p$, there is a specific 2-form $\omega {|}_p$, designed to do exactly one thing: it takes (or “eats”) the two vectors $v_1,v_2$ and produces a scalar, the exact flux of the wind through the patch of grass at that point.

All together:

$${\int }_S\omega =\underset{\text{partition refines}}{\lim }\sum _{\text{all elements}}\underset{\text{flux at point }p}{\underbrace{\omega {|}_p[[v_1,v_2]]}}$$

In words, we are taking the region $S$ and chopping it into an infinitesimal mesh. At each point $p$ in the mesh, there are two tangent vectors $v_1,v_2$ that span the local grid. Then we activate the sensor field $\omega$ at that point, giving us a scalar reading $\omega {|}_p[[v_1,v_2]]$. Finally, we sum up all these readings and take the limit as the mesh gets finer and finer, giving us the total flux of the wind through the entire patch $S$.

More generally, we can abstract integral over surfaces up to further dimensions and other types of $k$-forms.

Remark (Integration in 3D)

Following our 3D examples, this integration precisely reconstructs the classical vector integrals:

$$\begin{array}{rl}f(p)& ={\int }_p{f}_{0-\text{form}}\\ {\int }_{\Gamma }\mathbf{u}\cdot d\mathbf{l}& ={\int }_{\Gamma }{\mathbf{u}}_{1-\text{form}}\\ {\int }_S\mathbf{w}\cdot \mathbf{n}dA& ={\int }_S{\mathbf{w}}_{2-\text{form}}\\ {\int }_{V}fdV& ={\int }_V{f}_{3-\text{form}}\end{array}$$

Definition (Change of Coordinates and Pullback)

Let $S$ be a $k$-dimensional surface parameterized by a domain $D$ as $S=\varphi (D)$. Changing coordinates yields:

$${\int }_S\omega =\int \cdots {\int }_D{\omega }_{\varphi (\mathbf{x})}[[{\varphi }_{\ast }{e}_1,\dots ,{\varphi }_{\ast }{e}_k]]d{x}_1\dots d{x}_k$$

This structure mirrors the pullback operation from $M\to N$!

Definition (Pullback)

Given a linear map $A:U\to V$, the pullback for $k$-forms acts like an adjoint, pulling forms back from $V$ to $U$:

$$A^{\ast }:\stackrel{k}{\bigwedge }{V}^{\ast }\stackrel{\text{linear}}{\to }\stackrel{k}{\bigwedge }{U}^{\ast }$$

Defined by:

$$(A^{\ast }\omega )[[X_1,\dots ,X_k]]:=\omega [[A{X}_1,\dots ,A{X}_k]]$$

For a nonlinear map $\varphi :M\to N$, we have a canonical pullback for $k$-form fields (acting conceptually like back-propagation):

$${\varphi }^{\ast }:{\Omega }^k(N)\stackrel{\text{linear}}{\to }{\Omega }^k(M)$$

Defined point-wise as:

$$({\varphi }^{\ast }\omega {)}_p[[X_1,\dots ,X_k]]:={\omega }_{\varphi (p)}[[{\varphi }_{\ast }{X}_1,\dots ,{\varphi }_{\ast }{X}_k]]$$

where ${\varphi }_{\ast }$ is the pushforward (differential) of $\varphi$.

Theorem (Change of Coordinates using Pullback)

Using the pullback, the change of coordinates formula simplifies beautifully to:

$${\int }_{\varphi (D)}\omega ={\int }_D{\varphi }^{\ast }\omega$$

Example (Pullback in 3D Coordinates)

If we define the Jacobian matrix of the transformation $\varphi$ as $\mathbf{F}:=d\varphi$ and let $J:=\det \mathbf{F}$, the pullback operator maps our 3D vector forms:

$$\begin{array}{rl}{\varphi }^{\ast }{f}_{0-\text{form}}& =f\circ \varphi \\ {\varphi }^{\ast }({\mathbf{v}}_{1-\text{form}})& =({\mathbf{F}}^T(\mathbf{v}\circ \varphi ){)}_{1-\text{form}}\\ {\varphi }^{\ast }({\mathbf{w}}_{2-\text{form}})& =(J{\mathbf{F}}^{-1}(\mathbf{w}\circ \varphi ){)}_{2-\text{form}}\\ {\varphi }^{\ast }(f_{3-\text{form}})& =(J(f\circ \varphi ){)}_{3-\text{form}}\end{array}$$

Lemma (Homomorphism of Pullback)

The pullback is a homomorphism on the entire Exterior Algebra. It distributes over the wedge product:

$${\varphi }^{\ast }(\alpha \wedge \beta )=({\varphi }^{\ast }\alpha )\wedge ({\varphi }^{\ast }\beta )$$

It also interacts cleanly with the interior product (acting as insertion):

$$i_X({\varphi }^{\ast }\omega )={\varphi }^{\ast }(i_{{\varphi }_{\ast }X}\omega )$$

Definition (Exterior Derivative)

The exterior derivative $d$ is an operator that generalizes the concept of taking a differential, mapping $k$-form fields to $(k+1)$-form fields:

$$d:{\Omega }^k(M)\to {\Omega }^{k+1}(M)$$

The exterior derivative is defined by three rules:

1. For a scalar function $f\in {\Omega }^0(M)$, $df\in {\Omega }^1(M)$ is the standard differential of the function.
2. Nilpotent: $d(d\omega )=0$ (often stated simply as $d^2=0$).
3. Exterior Leibniz rule:

$$d(\alpha \wedge \beta )=(d\alpha )\wedge \beta +(-1{)}^{\text{deg}(\alpha )}\alpha \wedge (d\beta )$$

Example (Exterior Derivative in 3D Space)

1. $d{f}_{0-\text{form}}=(\nabla f{)}_{1-\text{form}}$ (Gradient)
2. $d{\mathbf{v}}_{1-\text{form}}=(\nabla \times \mathbf{v}{)}_{2-\text{form}}$ (Curl)
3. $d{\mathbf{w}}_{2-\text{form}}=(\nabla \cdot \mathbf{w}{)}_{3-\text{form}}$ (Divergence)

Lemma (Pullback and Exterior Derivative Commute)

The pullback operator and the exterior derivative commute:

$${\varphi }^{\ast }(d\omega )=d({\varphi }^{\ast }\omega )$$

Note

This commutativity is a profound result that is often proven directly using the generalized Stokes’ theorem. When translated into 3D coordinates using the mappings above, this single algebraic identity generates a host of non-obvious vector calculus identities!)

Theorem (Generalized Stokes’ Theorem)

The generalized Stokes Theorem connects integration of differential forms to the exterior derivative:

$${\int }_{S}d\omega ={\oint }_{\partial S}\omega$$

In 3D, this single, elegant theorem perfectly unifies three classical theorems of vector calculus into one single mathematical identity:

1. Fundamental Theorem of Calculus (0-forms): ${\int }_a^b\nabla f\cdot d\mathbf{l}=f(b)-f(a)$
2. Kelvin-Stokes Curl Theorem (1-forms): ${\iint }_S(\nabla \times \mathbf{u})\cdot \mathbf{n}dA={\oint }_{\partial S}\mathbf{u}\cdot d\mathbf{l}$
3. Gauss Divergence Theorem (2-forms): ${\iiint }_V(\nabla \cdot \mathbf{w})dV={\iint }_{\partial V}\mathbf{w}\cdot \mathbf{n}dA$