Differential Forms

Prerequisites: Vector Calculus, Manifold. Full path (4 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

A differential form is a machine that eats small arrows (or small patches of area, or small volumes) and returns a number, in a way that can be added up along a curve, over a surface or through a volume.

The simplest picture is a 1-form as a stack of surfaces, like the contour lines on a hiking map drawn in 3D. Given a small step, the 1-form returns the number of surfaces the step crosses, with a sign that says whether you cross them upwards or downwards. If the surfaces are the contour surfaces of a function, for example the height, then adding up the crossings along a path gives the change in height between start and end. That 1-form is the differential of the function. Not every 1-form comes from a function: the surfaces can end along lines, and then the total along a closed path need not vanish.

A 2-form does the same for small oriented patches of area: it returns something like the flux of a field through the patch, with a sign that depends on which side of the patch counts as "out". A 3-form measures small volumes. The orientation is part of the data, which is why exchanging two edges of a patch flips the sign.

Forms are the natural objects to integrate, because their values on small pieces are defined without choosing coordinates. With them, the fundamental theorem of calculus and the theorems of Gauss, Green and Stokes become a single statement: integrating a form over the boundary of a region gives the same as integrating its derivative over the region. Maxwell's equations take a short form in this language, and Hamiltonian mechanics is built on a 2-form on phase space.

Concrete

Differential 1-forms (co-vectors) are functions (elements of dual vector-space) which map vectors to real numbers.

• For the basic idea, see http://jakobschwichtenberg.com/vectors-forms-p-vectors-p-forms-and-tensors/
• One of the best introductions can be found in “Geometrical methods of mathematical physics” by Bernard F. Schutz
• Manifolds and Differential Forms lecture notes by Reyer Sjamaar
• Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by John H. Hubbard and Barbara Burke Hubbard - Extremely student friendly, lots of margin notes that talk about the "soft" stuff that's so crucial to the actual practice of math. Reading just the margins jumps your mathematical maturity by 2 years.
• Another good introduction can be found in “Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts”(2021) by Tristan Needham

Abstract

Let $M$ be a smooth $n$-manifold. A differential $k$-form is a smooth section of the $k$-th exterior power of the cotangent bundle,

$$\Omega^k(M) = \Gamma\big(\Lambda^k T^* M\big),$$

so at each point it is an alternating multilinear map $(T_pM)^k \to \mathbb{R}$. In local coordinates

$$\omega = \frac{1}{k!}\, \omega_{\mu_1 \dots \mu_k}(x)\, dx^{\mu_1} \wedge \dots \wedge dx^{\mu_k}$$

with totally antisymmetric components. The exterior product makes $\Omega^\bullet(M) = \bigoplus_{k=0}^n \Omega^k(M)$ a graded commutative algebra, $\alpha \wedge \beta = (-1)^{kl} \beta \wedge \alpha$ for $\alpha \in \Omega^k$, $\beta \in \Omega^l$.

The exterior derivative $d : \Omega^k \to \Omega^{k+1}$ is the unique $\mathbb{R}$-linear map that equals the differential on functions, satisfies $d^2 = 0$ and obeys the graded Leibniz rule

$$d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^k \alpha \wedge d\beta .$$

This makes $(\Omega^\bullet(M), \wedge, d)$ a differential graded algebra, the de Rham complex, whose cohomology is a topological invariant of $M$. Forms can be pulled back along any smooth map $f : N \to M$, and $f^*$ commutes with both $\wedge$ and $d$. Vector fields act through the interior product $\iota_X$ and the Lie derivative, related by Cartan's formula

$$\mathcal{L}_X = d\, \iota_X + \iota_X\, d .$$

Integration. On an oriented $n$-manifold, an $n$-form with compact support can be integrated without any further structure: in an oriented chart $\int \omega = \int \omega_{1 \dots n}\, d^n x$, and the change-of-variables formula makes this independent of the chart. Stokes' theorem holds for every compactly supported $(n-1)$-form on an oriented manifold with boundary (with the induced boundary orientation):

$$\int_M d\omega = \int_{\partial M} \omega .$$

Additional structure. A metric and an orientation define the Hodge dual $\star : \Omega^k \to \Omega^{n-k}$, the codifferential and the Laplacian on forms. In electrodynamics on spacetime the field strength is a 2-form $F$ with $dF = 0$, and the inhomogeneous equations take the form $d \star F \propto \star J$, with sign and normalization fixed by the conventions in use. A symplectic form on phase space is a closed, non-degenerate 2-form (symplectic structure). Forms with values in a vector bundle or a Lie algebra are used for connections and curvature.

Why is it interesting?

Path to this page

1. Calculus
2. Linear Algebra
3. Manifold
4. Vector Calculus

Leads to: Bianchi Identities, Chern-Simons Term, Cohomology, Exterior Derivative, Symplectic Structure.