2026-10-01 · chapter Yang–Mills and the mass gapPart 1 Mathematical preparationexterior derivative, Stokes, de Rham

Differential forms and the exterior derivative — gathering gradient, curl and divergence into one d, and counting holes

The $P\,dx + Q\,dy$ of a line integral looks like decoration inside the integral sign. Differential forms are the language that makes this "thing that is integrated" an object in its own right. How it changes under a change of coordinates is fixed by the chain rule, and gradient, curl and divergence gather into a single operation $d$.

The rule "replace $dx$ by $x'(t)\,dt$" is the first example of what §02 calls the pullback. Example. Traversing in the opposite direction changes the sign of (1), while the length $\int|\gamma'(t)|\,dt$ does not change. What differential forms measure are oriented quantities. The integrand of (1) is a rule that returns a number for the velocity $(x'(t), y'(t))$ at each point of the curve. This view, "returning a number for a tangent vector", is the starting point of §02.

In this book the electromagnetic field appears as $F = dA$ (§09), the gauge field as a matrix-valued 1-form (1-07), and the instanton number as the integral of a 4-form (3-05).

A linear function that returns a number for a tangent vector at a point $p$ (1-05) is called a cotangent vector, and a smooth assignment of one to each point is called a 1-form. The basic example is the differential of a function $df$, which returns the directional derivative for $v$ (repeated indices are summed).

$dx^i$ is the dual basis of $\partial/\partial x^j$, and under a change of coordinates the chain rule gives $dx'^j = \frac{\partial x'^j}{\partial x^i}\,dx^i$. The components $\alpha_i$ change by the transpose of the inverse of the matrix $\partial x'^j/\partial x^i$ that changes the components of tangent vectors, so $\alpha(v) = \alpha_i v^i$ does not depend on coordinates. The line integral along a curve $\gamma$ is the one-variable integral $\int_a^b\gamma^*\alpha$ of the pullback $\gamma^*\alpha = \alpha_i(\gamma(t))\,\dot\gamma^i(t)\,dt$.

Example 1. Pulling back $\alpha = -y\,dx + x\,dy$ along the unit circle $(\cos t, \sin t)$ gives $\gamma^*\alpha = dt$, and the integral is $2\pi$. Example 2. Since $\gamma^*df = \frac{d}{dt}f(\gamma(t))\,dt$,

and for a closed curve it is 0. Hence the $\alpha$ of Example 1 is not the $df$ of any $f$. But α is not "closed" in the sense of §07 either ($d\alpha = 2\,dx\wedge dy \ne 0$ by (7) of §04). An example that is closed yet not of the form $df$ is the $d\theta$ of §07.

The rule of pullback is the same even when it is not a curve. For a map $\varphi : u\mapsto x = \varphi(u)$, $\varphi^*\omega$ is ω with $x$ replaced by $\varphi(u)$ and $dx^i$ by $\frac{\partial\varphi^i}{\partial u^j}\,du^j$. A curve is the case where $u = t$ has one variable, and every $L^*$, $\varphi^*$ from §03 on has this meaning.

A 2-form returns a number from two tangent vectors, is linear in each, and changes sign when they are swapped. As a metaphor, "an area meter that remembers orientation"; to be exact, a smooth arrangement of antisymmetric bilinear forms on the tangent space at each point.

On the left is the wedge product of 1-forms, with the rule $\alpha\wedge\beta = -\beta\wedge\alpha$, in particular $\alpha\wedge\alpha = 0$. On the right is the oriented area of the parallelogram spanned by u, v, negative when v lies on the clockwise side of u (Figure 1). A k-form in n dimensions is a sum of $dx^{i_1}\wedge\cdots\wedge dx^{i_k}$ ($i_1 \lt \cdots \lt i_k$) with function coefficients; it has $\binom{n}{k}$ components, and is 0 if k > n. The wedge product of a k-form and an l-form is a (k+l)-form, and $\alpha\wedge\beta = (-1)^{kl}\,\beta\wedge\alpha$.

Example. In $\mathbb R^3$ both 1-forms and 2-forms have 3 components. Reading the wedge product of $a = a_i\,dx^i$ and $b = b_i\,dx^i$ through the coefficients of $dy\wedge dz,\ dz\wedge dx,\ dx\wedge dy$ gives the cross product $a\times b$ (agreement for 10⁴ random pairs Computed). A 2-form on $\mathbb R^4$ has 6 components, exactly enough for the 3 of the electric field and the 3 of the magnetic field (§09). Pulling back by a linear map $L$ gives $L^*(dx\wedge dy) = (\det L)\,dx\wedge dy$.

We extend $f \mapsto df$ to an operation from k-forms to (k+1)-forms (the exterior derivative in its present form is said to begin with É. Cartan (1899)). The rule is just "differentiate the coefficients, and multiply the resulting $dx^j$ from the left by the wedge product".

$d$ is linear, and for a k-form α, $d(\alpha\wedge\beta) = d\alpha\wedge\beta + (-1)^k\alpha\wedge d\beta$. And

The reason: $d(df) = \partial_i\partial_j f\,dx^i\wedge dx^j$, where the coefficients are symmetric in i, j (partial derivatives can be swapped) and $dx^i\wedge dx^j$ is antisymmetric, so the sum vanishes. For k-forms too it vanishes in the same way, by the symmetry of second derivatives of the coefficients and the antisymmetry of the wedge product. Example. In the plane, using only $dx\wedge dx = 0$ and $dy\wedge dx = -dx\wedge dy$,

KnownIn $\mathbb R^3$, the $d$ of a 0-form is the gradient, the $d$ of a 1-form $a_i\,dx^i$ is the curl ($(\partial_y a_z - \partial_z a_y)\,dy\wedge dz + \cdots$), and the $d$ of a 2-form $b_x\,dy\wedge dz + \cdots$ is the divergence $(\nabla\cdot\mathbf b)\,dx\wedge dy\wedge dz$. (6) states $\nabla\times\nabla f = 0$ and $\nabla\cdot(\nabla\times\mathbf a) = 0$ at once, and is the identity used when 2-03 §02 introduced potentials.

Since the pullback of §02 and $d$ can be interchanged ($\varphi^*d\omega = d\varphi^*\omega$) Known, $d$ does not depend on coordinates, and is defined on a manifold without a metric. The $df = \partial_r f\,dr + \partial_\theta f\,d\theta$ in polar coordinates has no $1/r$ of the gradient formula; $1/r$ comes in from the metric.

A k-form is integrated over an oriented k-dimensional piece. Describe the piece by $\varphi : D\subset\mathbb R^k \to M$, and pull back to an ordinary multiple integral: $\int_\varphi\omega = \int_D\varphi^*\omega$. From (4) follows:

In place of the $|\det|$ of the change-of-variables formula, the signed $\det$ appears, and reversing the orientation changes the sign. On a whole manifold we compute chart by chart and glue, so the orientation must be consistent over the whole (orientable) (the Möbius strip is not) Known.

Example. In polar coordinates $dx\wedge dy = (\cos\theta\,dr - r\sin\theta\,d\theta)\wedge(\sin\theta\,dr + r\cos\theta\,d\theta) = r\,dr\wedge d\theta$. The $r$ of the area element comes from the wedge product.

Example (a surface). Orient the upper hemisphere outwards and describe it in spherical coordinates $\psi(\vartheta, \varphi) = (\sin\vartheta\cos\varphi, \sin\vartheta\sin\varphi, \cos\vartheta)$ ($0 \le \vartheta \le \pi/2$). The order $(\vartheta, \varphi)$ matches the outward orientation. Computing the $\det$ of (8) (the 2×2 for the two components $x, y$) gives $\psi^*(dx\wedge dy) = \cos\vartheta\sin\vartheta\,d\vartheta\wedge d\varphi$, so

(numerical integration and a check of the orientation Computed). The value is the same as the volume of the solid hemisphere. This is not a coincidence but an instance of Stokes' theorem in the next section, §06: $d(z\,dx\wedge dy) = dx\wedge dy\wedge dz$, and on the base disc, part of the boundary of the solid hemisphere, $z = 0$.

"The total of the derivative inside is the total of the values on the boundary." It is the form of (3) gathered for all dimensions. If M is a compact oriented n-dimensional manifold (it may have a boundary $\partial M$) and ω an (n−1)-form, then

The orientation of $\partial M$ is taken so that putting the outward normal first gives the orientation of M. For an interval it is (3); for a planar region it is Green's theorem via (7); for a surface in $\mathbb R^3$ it is Stokes' theorem for the curl; for a solid it is Gauss's divergence theorem Known.

Sketch of the proof. On a rectangle $[a, b]\times[c, d]$, if $\omega = Q\,dy$ then $\iint\partial_x Q\,dx\,dy = \int_c^d(Q(b, y) - Q(a, y))\,dy$, which is the sum of the line integrals along the right edge (upwards) and the left edge (downwards). A general region is divided into small squares. Adjacent squares traverse their common edge in opposite directions, so the inner edges cancel and only the outer boundary remains. Meanwhile, the circulation around a small square approaches "$d\omega$ at the centre × area".

ComputedTable 1 For the ω of Figure 2, the circulation of each small square found as the sum of the line integrals along its four edges (no surface integrals used).

For every N the total agrees with the $\oint\omega = 6$ of the outer boundary (difference at most 3×10⁻¹⁵), and the deviation decreases exactly as $1/N^2$ (on a square of side $h = 2/N$ the average of $3y^2$ is larger than the value at the centre by $h^2/4$). On the unit disc too both sides agree at $5\pi/4$ Computed.

If $d\omega = 0$ the form is called closed, and if there is an η with $\omega = d\eta$ it is called exact. By (6), exact implies closed. The converse depends on the shape of the space. On the plane with the origin removed:

By (7), $\partial_x\frac{x}{r^2} + \partial_y\frac{y}{r^2} = \frac{2r^2 - 2x^2 - 2y^2}{r^4} = 0$, so it is closed. On the unit circle, as in Example 1, $\oint d\theta = 2\pi$, so by (3) it is not exact. The argument θ increases by 2π in one turn, so it is a function only on a region with a cut (for example the negative x axis). dθ is defined without a cut.

The Poincaré lemma. On a star-shaped region (one containing the segment joining some point to each point) every closed form is exact Known. If the region is star-shaped with respect to the origin, then for a closed 1-form α

satisfies $d\eta = \alpha$ (checked with $d(\sin x\,e^y + x^3y)$; moving the base point to (1, 0), θ is recovered from dθ on the right half-plane Computed).

The winding number. $\frac{1}{2\pi}\oint_\gamma d\theta$ is the number of times the closed curve γ goes round the origin, and does not change under deformations that do not cross the origin (dθ is closed on the band swept by the deformation, so by (9) the difference is 0). θ jumps from π to −π each time it crosses the negative x axis from above to below, so the winding number is also the oriented number of crossings of the cut.

ComputedTable 2 $\frac{1}{2\pi}\oint d\theta$. Computed in two ways at 2×10⁵ points; the difference from an integer is below 10⁻⁹.

The index of the zeros in the detour of 1-05 (the hairy ball theorem) is also this winding number. Outside the solenoid of 2-03 §06, $A = \frac{\Phi}{2\pi}\,d\theta$, and although $dA = 0$ the circulation is Φ. The Aharonov–Bohm effect is the physics of a form that is closed but not exact.

Among closed forms, regard the exact ones as 0, and count what remains.

$b_k$ is called a Betti number. Example. $H^0$ consists of functions constant on each connected component, so $b_0$ is the number of connected components. $H^1(\mathbb R^2\setminus\{0\})$ is the one dimension spanned by [dθ] (subtracting $c\,d\theta$ from a closed form ω so that the circulation on the unit circle is 0 makes it exact. Every closed curve can be deformed, without crossing the origin, into "a curve that goes round the unit circle some number of times", and deformation does not change circulation (the argument of §07), so every circulation is 0. Then the line integral from a base point does not depend on the path and defines a function Known). The area form σ of the sphere is closed and $\int_{S^2}\sigma = 4\pi$, but if $\sigma = d\eta$, then since $S^2$ has no boundary, (9) would make the integral 0. Hence it is not exact.

De Rham's theorem (1931), in today's language: $H^k$ agrees with cohomology (with real coefficients) defined from topology alone, without forms Known. Examples: $\mathbb R^n$ is (1, 0, 0, …), $\mathbb R^3\setminus\{0\}$ is (1, 0, 1, 0), $S^4$ is (1, 0, 0, 0, 1) (the stage of 3-05).

The point magnetic charge. The 2-form $\beta = (x\,dy\wedge dz + y\,dz\wedge dx + z\,dx\wedge dy)/r^3$ on $\mathbb R^3\setminus\{0\}$ is the magnetic field $\hat{\mathbf r}/r^2$, and $d\beta = 0$, but its integral over a closed surface enclosing the origin is 4π (difference below 10⁻⁹ for a sphere and a shifted cube, and 0 for a sphere not enclosing it Computed). Hence there is no A with $dA = \beta$ on the whole space. Off the negative z axis $A_N = (1 - \cos\vartheta)\,d\varphi$ can be used, and off the positive z axis $A_S = -(1 + \cos\vartheta)\,d\varphi$ (spherical coordinates); the difference $2\,d\varphi$ is a gauge transformation by the multivalued function 2φ Computed. Taking potentials separately on the two regions, north and south, and gluing them on the overlap by a gauge transformation is in §III of Wu–Yang (1975). The language of this gluing is bundles and connections (1-07).

If the total magnetic flux is Φ, the difference is $\frac{\Phi}{2\pi}\,d\varphi$ (here Φ = 4π). The phase factor $e^{iq\chi/\hbar}$ of a particle of charge q (2-03) is single-valued under $\chi = \frac{\Phi}{2\pi}\varphi$ when $q\Phi/h$ is an integer. The quantisation of the point magnetic charge is due to Dirac (1931), and the reading in the language of bundles is in 1-07 §08.

Detour: differential forms on a lattice

Up to here we have not used a metric. The Hodge dual needs one, so in this section we define on the spot only as much as we use. A metric is given by the table of inner products $\eta_{ij}$ between coordinate directions (here diagonal, with entries ±1). In Minkowski spacetime (2-04), for coordinates $(t, x, y, z)$, $\eta = \operatorname{diag}(-1, 1, 1, 1)$ (signature $(-,+,+,+)$, $c = \varepsilon_0 = \mu_0 = 1$); in Euclidean $\mathbb R^4$, $\eta = \operatorname{diag}(1, 1, 1, 1)$.

• Volume form: $\mathrm{vol} = dt\wedge dx\wedge dy\wedge dz$ (in the Euclidean case $dx^1\wedge dx^2\wedge dx^3\wedge dx^4$).
• Inner product of forms: for basis elements $\langle dx^{i_1}\wedge\cdots\wedge dx^{i_k},\ dx^{i_1}\wedge\cdots\wedge dx^{i_k}\rangle = \eta_{i_1i_1}\cdots\eta_{i_ki_k}$, and basis elements with different sets of indices give 0. For general forms extend bilinearly in the coefficients.
• Hodge dual: the linear map taking a k-form β to an (n−k)-form $*\beta$ such that $\alpha\wedge *\beta = \langle\alpha, \beta\rangle\,\mathrm{vol}$ for every k-form α.

Example. In Minkowski, from $\langle dt\wedge dx, dt\wedge dx\rangle = \eta_{tt}\eta_{xx} = -1$ and $dt\wedge dx\wedge dy\wedge dz = \mathrm{vol}$, $*(dt\wedge dx) = -dy\wedge dz$. From $\langle dy\wedge dz, dy\wedge dz\rangle = +1$ and $dy\wedge dz\wedge dt\wedge dx = \mathrm{vol}$, $*(dy\wedge dz) = dt\wedge dx$ (checked for all 6 basis 2-forms Computed). A −1 enters only for basis elements containing time, and this affects the sign in (14) and the sign of $**$ below.

We use it for the electromagnetic field. Gathering the potentials into a 1-form gives

($\mathbf E = -\nabla\phi - \partial_t\mathbf A$, $\mathbf B = \nabla\times\mathbf A$). The four Maxwell equations become two.

$dF = 0$ is automatic from (6) and is the two equations (M2) of 2-03; $d{*F} = {*j}$ is the two equations (M1) (checked with random potentials Computed). That F does not change under the gauge transformation $A \mapsto A + d\chi$ is also (6) ((G) of 2-03). What enters the action is $F\wedge *F = (|\mathbf B|^2 - |\mathbf E|^2)\,\mathrm{vol}$, and also $F\wedge F = -2(\mathbf E\cdot\mathbf B)\,\mathrm{vol}$ Computed.

What happens only in four dimensions

$*F$ is an (n−2)-form, and has the same degree as F only for n = 4. The $**$ on 2-forms is +1 in Euclidean $\mathbb R^4$ (the eigenvalues ±1 of $*$ three each) and −1 in Minkowski (eigenvalues $\pm i$) (agreeing with the formula $(-1)^{k(n-k)+s}$, where s is the number of negative signs, for every k Computed). In the Euclidean case it splits as $F = F^+ + F^-$ (self-dual and anti-self-dual), and $|F|^2 = |F^+|^2 + |F^-|^2 \ge \bigl||F^+|^2 - |F^-|^2\bigr| = |\langle F, *F\rangle|$. Integrating gives $\int F\wedge *F \ge |\int F\wedge F|$, with equality when $F = \pm *F$; this is the skeleton of the instanton inequality (3-05).

In Minkowski, $** = -1$ on 2-forms, so if $*F = \lambda F$ then $-F = \lambda^2F$, and the only real F satisfying $*F = \pm F$ is 0. In physics, instantons are considered in imaginary time (2-04, 4-02). For U(1), $F\wedge F = d(A\wedge F)$ is exact, so its integral is fixed by the boundary term (at infinity) alone. In the non-abelian case this becomes an integer-valued topological quantity (3-05).

See also the article on four dimensions and 1-12. In the non-abelian case $F = dA + A\wedge A$ ($A$ is the geometric way of writing with values in anti-Hermitian matrices; the translation to the physicists' way is in 3-02), and in place of $dF = 0$ we have $dF + A\wedge F - F\wedge A = 0$ (Bianchi) (1-07, 3-03).

What this chapter can say

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