Differential Forms and Stokes’ Theorem

Integration on a manifold needs something to integrate that doesn't depend on the coordinates. Functions won't do: their integrals change with the coordinates by the Jacobian factor (3A.5 Product Measures and Change of Variables). The right objects are differential forms, alternating tensors that transform by exactly the determinant, so their integrals are coordinate-free. Forms have a natural derivative, the exterior derivative $d$, and the relation between the two is Stokes' theorem:

It contains the fundamental theorem of calculus, Green's theorem, the divergence theorem and the classical Stokes theorem as special cases (1A.10 Divergence, Curl and the Integral Theorems).

For the Ricci flow its most important consequence is integration by parts on a closed manifold: with no boundary, $\int_M\Delta u\,dV = 0$ and $\int_Mf\,\Delta u = -\int_M\langle\nabla f, \nabla u\rangle$. Every monotonicity formula in the subject, from the energy decay of 6A.3 The Heat Equation on ℝⁿ to Perelman's $\mathcal F$ and $\mathcal W$, is proved by differentiating an integral and integrating by parts until the derivative is visibly a sum of squares. This chapter supplies that tool.

By the end of this chapter you will be able to:

• define differential forms, the wedge product, the exterior derivative and pullback, and compute with them in coordinates;
• explain orientation and define the integral of a form on an oriented manifold;
• state Stokes' theorem and derive Green's theorem and the divergence theorem from it;
• integrate by parts on a closed Riemannian manifold, and show $\int\Delta u = 0$;
• distinguish closed and exact forms, with the example $d\theta$ on the punctured plane.

8.1Measuring area by tracing the boundary

8.2Forms and the exterior derivative

A $k$-form at a point is an alternating $(0, k)$-tensor (8A.7 Tensors and Index Notation): a multilinear function of $k$ tangent vectors that changes sign when two are swapped. A differential $k$-form on $M$ is a smooth field of them. In coordinates, the basic $k$-forms are the wedge products $dx^{i_1}\wedge\dots\wedge dx^{i_k}$ with $i_1 < \dots < i_k$, and every $k$-form is

The wedge product is bilinear, associative and graded-commutative: $\alpha\wedge\beta = (-1)^{kl}\beta\wedge\alpha$ for a $k$-form and an $l$-form; in particular $dx\wedge dx = 0$ and $dx\wedge dy = -dy\wedge dx$. On an $n$-manifold, $n$-forms are multiples of $dx^1\wedge\dots\wedge dx^n$, and forms of degree $> n$ vanish.

The value of $dx\wedge dy$ on two vectors $v$, $w$ in the plane is $v^1w^2 - v^2w^1$, the signed area of the parallelogram they span (1A.3 Determinants and Volume). A $k$-form measures oriented $k$-dimensional volume.

It is independent of coordinates, and satisfies

for every smooth map $F$, where $F^*$ is the pullback of forms (8A.6 Flows and the Lie Derivative). The identity $d^2 = 0$ is the symmetry of second derivatives, $\partial_i\partial_jf = \partial_j\partial_if$, combined with $dx^i\wedge dx^j = -dx^j\wedge dx^i$. In $\mathbb{R}^3$, under the identifications of 1-forms and 2-forms with vector fields, $d$ on $0$-, $1$- and $2$-forms is the gradient, the curl and the divergence, and $d^2 = 0$ is the pair of identities $\operatorname{curl}\operatorname{grad} = 0$ and $\operatorname{div}\operatorname{curl} = 0$ (Exercise 8.4).

8.3Orientation and integration

An orientation of $M$ is a choice of charts whose transition maps all have positive Jacobian determinant; $M$ is orientable if one exists (7A.3 Manifolds and Surfaces). The Möbius band and $\mathbb{R}P^2$ are not orientable; spheres, tori and all Lie groups are. Equivalently, $M$ is orientable if and only if it has a nowhere-vanishing $n$-form.

For an $n$-form $\omega = f\,dx^1\wedge\dots\wedge dx^n$ supported in the domain of an oriented chart, define

This is the reason for using forms: under a change of oriented charts, $f$ is multiplied by the Jacobian determinant (Exercise 8.5), exactly the factor in the change of variables formula (3A.5 Product Measures and Change of Variables), so the integral is independent of the chart. In general, write $\omega = \sum\psi_\alpha\omega$ with a partition of unity and add the pieces; the result doesn't depend on the choices (8A.2 Partitions of Unity, last exercise).

A Riemannian metric gives an oriented manifold a canonical $n$-form, the volume form

in oriented coordinates, the form that gives an orthonormal basis volume $1$. Functions can then be integrated: $\int_Mf\,dV_g$. On a non-orientable manifold one integrates densities instead, with the same formula and $|{\det}|$; all the integrals of this guide make sense on any manifold.

8.4Stokes' theorem

A manifold with boundary (7A.3 Manifolds and Surfaces) has an induced orientation on its boundary: the outward normal first convention. In the plane it means the boundary of a region is traversed counterclockwise, keeping the region on the left (Figure 8.2).

Special cases. For a $0$-form on an interval, Stokes is the fundamental theorem of calculus, $\int_a^bf' = f(b) - f(a)$. For a 1-form $P\,dx + Q\,dy$ on a plane region it is Green's theorem (Exercise 8.6). For a 2-form on a surface in $\mathbb{R}^3$ it is the classical Stokes theorem relating the flux of a curl to a circulation, and for a 2-form on a region of $\mathbb{R}^3$ it is the divergence theorem (1A.10 Divergence, Curl and the Integral Theorems). In physics, Maxwell's equations take the form $dF = 0$ and $d{\star}F = J$ for the electromagnetic field 2-form $F$ on spacetime, and Gauss's law and Faraday's law in integral form are Stokes' theorem applied to them. The circulation of an ideal flow (5A.2 Cauchy’s Theorem and Its Consequences) and the lift on a wing (5A.1 Holomorphic Functions Are Conformal) are line integrals that Stokes relates to the vorticity inside.

8.5Divergence and integration by parts

On a Riemannian manifold the divergence of a vector field $X$ is the function with $d(\iota_XdV) = (\operatorname{div}X)\,dV$, where $\iota_X$ inserts $X$ as the first argument; in coordinates

and in terms of the covariant derivative of 9A.2 Connections, $\operatorname{div}X = \nabla_iX^i$. It measures the rate at which the flow of $X$ changes volume: $\mathcal L_XdV = (\operatorname{div}X)\,dV$, by Cartan's formula (8A.6 Flows and the Lie Derivative). The Laplacian of a function is $\Delta u = \operatorname{div}\nabla u$.

These identities are what make energy methods work on closed manifolds: the Laplacian is a negative semidefinite symmetric operator, $\int u\Delta u = -\int|\nabla u|^2 \leq 0$, with the constants as its kernel (6A.5 Weak Solutions and Elliptic Regularity, 9A.6 The Laplacian and the Bochner Formula). On a closed manifold, the heat equation conserves $\int u$, decreases $\int u^2$, and the maximum principle needs no boundary conditions (6A.4 Maximum Principles, 9B.7 The Heat Equation on a Manifold).

8.6Closed and exact forms

A form is closed if $d\omega = 0$, and exact if $\omega = d\eta$ for some $\eta$. Since $d^2 = 0$, exact forms are closed. On $\mathbb{R}^n$, or any star-shaped region, every closed form is exact (the Poincaré lemma). On the punctured plane the form

is closed but not exact: its integral around the unit circle is $2\pi$, while the integral of an exact form $df$ around any closed curve is $0$ by Stokes. The notation $d\theta$ is a warning, not a contradiction: the angle $\theta$ exists only locally (7A.4 The Fundamental Group, 5A.2 Cauchy’s Theorem and Its Consequences). The de Rham cohomology groups $H^k_{\mathrm{dR}}(M) = \{\text{closed }k\text{-forms}\}/\{\text{exact }k\text{-forms}\}$ measure the failure of closed forms to be exact; de Rham's theorem identifies them with the homology of 7A.8 Homology in Brief, over $\mathbb{R}$. A closed 1-form whose integral around every loop vanishes is exact, so $H^1_{\mathrm{dR}}(M) = 0$ on a simply connected manifold: the form $d\theta$ exists on the punctured plane because a loop goes around the hole.

8.7History

Differential forms were introduced by Élie Cartan in 1899, building on Hermann Grassmann's exterior algebra (1844). The general Stokes theorem took its modern form in Cartan's work and was stated for manifolds in the 1930s; its classical cases are due to Green (1828), Gauss and Ostrogradsky (divergence theorem, 1813–1831) and Kelvin and Stokes (1850–54). Georges de Rham proved his theorem in 1931. Jakob Amsler invented the polar planimeter in 1854.

8.8Exercises

Solution

$d(P\,dx + Q\,dy) = P_y\,dy\wedge dx + Q_x\,dx\wedge dy = (Q_x - P_y)\,dx\wedge dy$. For the area, take $P = -\frac y2$, $Q = \frac x2$, so $Q_x - P_y = 1$.

Solution

$x\,dy - y\,dx = ab(\cos^2t + \sin^2t)\,dt$, so the area is $\frac12\int_0^{2\pi}ab\,dt = \pi ab$. If $d\omega_1 = d\omega_2 = dx\wedge dy$, then $d(\omega_1 - \omega_2) = 0$, and by Stokes $\oint_{\partial D}(\omega_1 - \omega_2) = \iint_Dd(\omega_1 - \omega_2) = 0$.

Solution

(a) $\Delta(e^{-f}) = \operatorname{div}(-e^{-f}\nabla f) = e^{-f}|\nabla f|^2 - e^{-f}\Delta f$, and its integral is $0$. (b) $\operatorname{div}(u\,e^{-f}\nabla v) = e^{-f}\langle\nabla u, \nabla v\rangle + u\,e^{-f}\Delta v - u\,e^{-f}\langle\nabla f, \nabla v\rangle$, and its integral vanishes.