Module 02 — Differential Forms Refresher
Why this module: The whole payoff of this study compresses into two equations of forms: $F = dA$ and $dF = 0$. Connections are Lie-algebra-valued 1-forms; curvature is a 2-form. If forms are rusty, everything after Module 04 will feel like symbol-pushing.
Core idea
A $k$-form is an antisymmetric multilinear machine eating $k$ tangent vectors, varying smoothly over the manifold. The Exterior Derivative $d$ generalizes grad/curl/div at once, and $d^2 = 0$ is "curl grad = 0" and "div curl = 0" — and, deeper, the source of de Rham Cohomology.
Physics anchor: Landau's antisymmetric field tensor $F_{ik}$ (§23) is a 2-form in coordinate clothing. The first pair of Maxwell equations (§26) is exactly $dF = 0$.
Concepts to (re)work
- Differential Form — wedge product, coordinate expressions
- Exterior Derivative — definition, $d^2=0$, naturality ($d$ commutes with pullback)
- Pushforward and Pullback — forms pull back; this is why they're better than vectors
- Stokes Theorem — $\int_M d\omega = \int_{\partial M} \omega$, the master integral theorem
- de Rham Cohomology — closed vs. exact; measures global topology, needed for Aharonov-Bohm Effect and Dirac Monopole
Reading
- Sternberg - Lectures on Differential Geometry, Ch. I (exterior algebra) and Ch. III (integral calculus)
Self-check before moving on
Can you compute $d$ of any given form in coordinates? Explain why closed-but-not-exact forms detect holes?
Concepts in this module
Study tasks
- Read the five concept notes and annotate
- Exercise: write $E$ and $B$ fields as a single 2-form $F$ on $\mathbb{R}^4$ and check $dF=0 \Leftrightarrow$ (no monopoles + Faraday)
- Exercise: show $\omega = \frac{-y\,dx + x\,dy}{x^2+y^2}$ on $\mathbb{R}^2\setminus\{0\}$ is closed but not exact — keep this example; it *is* the Aharonov–Bohm effect
- Recompute Stokes theorem on a small example (disk, circle)