Differential Forms

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 $\omega = \sum_{i < j} \omega_{ij}\, dx^i \wedge dx^j$.

Definition. If $F : M \to N$ is a smooth map, the pullback $F^* : \Omega^k(N) \to \Omega^k(M)$ is defined by:

$(F^*\omega)_p(v_1, \ldots, v_k) = \omega_{F(p)}(dF_p(v_1), \ldots, dF_p(v_k))$

for $v_i \in T_p M$.

Proposition 4.1. The pullback satisfies:

1. $F^*(\alpha \wedge \beta) = F^*\alpha \wedge F^*\beta$
2. $F^*(d\omega) = d(F^*\omega)$
3. $(G \circ F)^* = F^* \circ G^*$

Definition. The interior product (contraction) of a vector field $X$ with a $k$-form $\omega$ is the $(k-1)$-form $\iota_X \omega$ defined by:

$(\iota_X \omega)(v_1, \ldots, v_{k-1}) = \omega(X, v_1, \ldots, v_{k-1})$

Proposition 4.2 (Cartan’s Magic Formula). The Lie derivative $\mathcal{L}_X$ of a differential form satisfies:

$\mathcal{L}_X \omega = d(\iota_X \omega) + \iota_X(d\omega)$

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 4.3. $d \circ d = 0$ ($d^2 = 0$).

Theorem 4.4 (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 $\eta$.

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

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

Theorem 4.5 (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 ($n = 3$): $\oint_{\partial V} \mathbf{F} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{F})\, dV$.

Integration of an $n$-form over an $n$-dimensional oriented manifold is defined by pulling back to $\mathbb{R}^n$ and integrating in coordinates. If $\omega = f\,dx^1 \wedge \cdots \wedge dx^n$ on a coordinate chart $U$ with parametrisation $\phi : U \to \mathbb{R}^n$:

$\int_U \omega = \int_{\phi(U)} f(x^1, \ldots, x^n)\, dx^1 \cdots dx^n$

Problem. Compute $\int_{S^2} \omega$ where $\omega = x\,dy \wedge dz + y\,dz \wedge dx + z\,dx \wedge dy$ on the unit sphere $S^2 \subset \mathbb{R}^3$.

An orientation on an $n$-dimensional manifold is a nowhere-vanishing $n$-form. A manifold is orientable if such a form exists. The Möbius strip is the classic example of a non-orientable manifold: any attempt to define a global $n$-form results in a sign change along the closed loop.

Integration of an $n$-form over an oriented manifold is independent of the choice of atlas (as long as the charts are orientation-preserving). The integral changes sign if the orientation is reversed.

Differential forms unify the classical vector calculus operators in $\mathbb{R}^3$ via the identifications:

• $0$-forms $\leftrightarrow$ scalar functions
• $1$-forms $\leftrightarrow$ vector fields (via $F_1 dx + F_2 dy + F_3 dz \leftrightarrow \mathbf{F}$)
• $2$-forms $\leftrightarrow$ vector fields (via $F_1 dy\wedge dz + F_2 dz\wedge dx + F_3 dx\wedge dy \leftrightarrow \mathbf{F}$)
• $3$-forms $\leftrightarrow$ scalar functions (via $f\, dx\wedge dy\wedge dz \leftrightarrow f$)

Under these identifications:

• $df \leftrightarrow \nabla f$ (gradient)
• $d\omega \leftrightarrow \nabla \times \mathbf{F}$ (curl) for a $1$-form $\omega$
• $d\omega \leftrightarrow \nabla \cdot \mathbf{F}$ (divergence) for a $2$-form $\omega$

The identity $d^2 = 0$ becomes $\nabla \times (\nabla f) = 0$ and $\nabla \cdot (\nabla \times \mathbf{F}) = 0$.