Differential Forms

4.1 Alternating Tensors

The space of $k$-covectors at $p$ is $\Lambda^k(T_p^* M)$, the space of alternating $k$-linear maps $T_p M \times \cdots \times T_p M \to \mathbb{R}$.

Definition. A differential $k$-form on $M$ is a smooth section of $\Lambda^k T^* M$, i.e., a smooth map $\omega : M \to \Lambda^k T^* M$ assigning to each $p$ an alternating $k$-linear functional $\omega_p$ on $T_p M$.

Examples.

• A $0$-form is a smooth function $f \in C^\infty(M)$.
• A $1$-form is a section of $T^* M$ (a covector field). In local coordinates, $\omega = \sum \omega_i\, dx^i$.
• A $2$-form on $M$ has the local expression $\omega = \sum_{i < j} \omega_{ij}\, dx^i \wedge dx^j$.

4.2 Exterior Derivative

The exterior derivative is the operator $d : \Omega^k(M) \to \Omega^{k+1}(M)$ defined by:

• For $f \in \Omega^0(M)$: $df = \sum_{i=1}^n \frac{\partial f}{\partial x^i}\, dx^i$ (the total differential).
• For general $k$-forms: defined by requiring $d(df) = 0$ and the product rule $d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^{\deg(\alpha)} \alpha \wedge d\beta$.

Proposition 4.1. $d \circ d = 0$ ($d^2 = 0$).

Theorem 4.2 (Poincare Lemma). If $M$ is a star-shaped open subset of $\mathbb{R}^n$ (or more generally, a contractible manifold), then every closed $k$-form is exact: if $d\omega = 0$, then $\omega = d\eta$ for some $(k - 1)$-form $\eta$.

4.3 Wedge Product

The wedge product $\wedge : \Omega^k(M) \times \Omega^\ell(M) \to \Omega^{k+\ell}(M)$ is the bilinear, associative, anti-commutative operation:

$\alpha \wedge \beta = (-1)^{k\ell}\, \beta \wedge \alpha$

4.4 Stokes” Theorem

Theorem 4.3 (Stokes’ Theorem). Let $M$ be an oriented $n$-dimensional manifold with boundary $\partial M$ (with the induced orientation). If $\omega$ is a compactly supported $(n-1)$-form on $M$, then:

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

Special Cases:

• Fundamental Theorem of Calculus ($n = 1$): $\int_a^b f'(x)\, dx = f(b) - f(a)$.
• Green’s Theorem ($n = 2$): $\oint_{\partial D} P\, dx + Q\, dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dx\, dy$.
• Classical Stokes’ Theorem ($n = 3$): $\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$.
• Divergence Theorem ($n = 3$): $\oint_{\partial V} \mathbf{F} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{F})\, dV$.