Chapter 5: What Does It Mean to Differentiate? — The Exterior Derivative $d$: Integral Quantities and Local Laws

§5.0 The Bridge to Differentiation — Observation Is Integral, Law Is Differential

In Chapter 3 we defined integration over curves, surfaces, and regions in the language of $k$-forms. Work along a curve $\gamma$ is $\int_\gamma \omega$, flux through a surface $S$ is $\iint_S \eta$, and the total mass of a region $V$ is $\iiint_V \Omega$. Each followed the same principle: feed $k$ displacement vectors to a $k$-form (measuring device), obtain a scalar, and aggregate over the entire region.

In Chapter 4 we introduced the operation that rebuilds measuring devices to compute the same integral in different variables—the pullback $\Phi^*$. With that, we also gained a tool for changing the variables of computation while preserving the integral value.

The integrals we now have in hand are all quantities that span an entire region (below we call such quantities global). Work is the total sum over an entire curve, flux is the total amount crossing an entire surface, and mass is the grand total over an entire region. They depend on how the measuring device is applied and on the choice of region.

But this alone is not what physicists ultimately want. Behind the laws observed as integral quantities, they want to see local relations that hold at each point. Whether Maxwell's equations or the Navier–Stokes equations, the fundamental laws of nature are written as local relations—differential equations—that describe what holds at each point in space and each instant in time. Global integral quantities depend on the region, but local laws have a universality that does not depend on the region.

Note (the terms global / local) Other books sometimes call a similar contrast macro / micro. In this chapter, however, we use global / local to mean the contrast between a quantity integrated over an entire region and a relation that holds at each point—not so much a difference of scale.

Here a fundamental question arises.

How do we extract a local law from an integrated quantity?

The subject of this chapter—the exterior derivative $d$—answers this question. $d$ is an operator that acts on a $k$-form and returns a $(k+1)$-form; combined with the integrals defined in Chapter 3, it translates a "global integral measured on the boundary" into "an accumulation of local changes in the interior." In short, $d$ is the device that converts integral laws into differential laws.

The structure of this chapter is as follows. First we recall the $df$ introduced in Chapter 1 and build $d\omega$ from the mismatch that remains when a general $1$-form is measured on a closed loop. Next we extend the same idea to $2$-forms and unify Stokes' theorem and Gauss' theorem into a single form. Finally we look at the structure $d^2=0$ and at how the exterior derivative localizes physical laws, connecting to the Hodge star in the next chapter.

§5.1 Revisiting $df$ — Are Differentiation and Integration Inverse Operations?

5.1.1 $df$ Is "Raising the Degree"

In Chapter 1 §1.2 we defined the total differential of a function $f(x,y,z)$ as a row vector:

This is an operation that takes a $0$-form (the scalar field $f$) as input and outputs a $1$-form (the row vector $df$). The degree rises from 0 to 1.

Recall here that $df(\mathbf{v})$ returns the first-order change in $f$ for a displacement $\mathbf{v}$. When $\mathbf{v} = \begin{pmatrix}\Delta x\\\Delta y\\\Delta z\end{pmatrix}$ is sufficiently small:

$df$ by itself is not "the change itself"—as agreed in Chapter 1 §1.1.6, $df$ is an operator; a numerical value is obtained only when it acts on a displacement vector.

5.1.2 A Line Integral Leaves Only the Difference of Endpoints

Integrate this $df$ along a curve $\gamma$. As in Chapter 3 §3.3.1, partition the curve into small intervals, feed $df$ each step's displacement $\Delta\mathbf{r}_i$, and add up the results.

To first order on each interval, $df(\Delta\mathbf{r}_i) \approx f(\mathbf{r}_i) - f(\mathbf{r}_{i-1})$, so the sum over all intervals is:

Expanding this sum:

Adjacent terms $f(\mathbf{r}_1), f(\mathbf{r}_2), \dots, f(\mathbf{r}_{n-1})$ cancel in plus and minus—a telescoping sum. Only the first and last survive. In the limit of infinitely fine partition:

where $A$ is the starting point of the curve and $B$ is the endpoint.

Checkpoint so far

- $\int_\gamma df = f(B) - f(A)$. The result of the integral depends entirely on the endpoint values, not at all on the shape of the path.

- This holds precisely because $df$ is a special $1$-form. In mechanics, it is the mathematical reason why "the work of a conservative force does not depend on the path."

5.1.3 Checking with a Chapter 3 Example

In Chapter 3 §3.4.1 we integrated $\omega = y\,dx + x\,dy$ along the unit circle $\gamma(t) = (\cos t,\; \sin t,\; 0),\; t \in [0,2\pi]$ and found the result to be $0$. At the time we integrated straightforwardly along the coefficients, but viewed in today's language the story is much simpler.

In fact $y\,dx + x\,dy$ is nothing other than $d(xy)$. For:

Therefore $\omega = df$ (with $f = xy$), and the telescoping-sum argument of §5.1.2 applies as-is. The unit circle is a closed curve, so $B = A$, and hence:

The calculation $\int_0^{2\pi} \cos 2t\,dt = 0$ from back then mechanically gave zero precisely because $\omega$ was an exact form, namely $d(xy)$.

Note (exact forms) A $1$-form that can be written as $\omega = df$ is called an exact form. Integrating an exact form along a closed curve always gives zero, as long as $f$ is a single-valued function. On the other hand, beware: "the integral along some closed curve was zero" does not necessarily mean the form is exact (whether the integral is zero on all closed curves also depends on the topology of the region). We examine this point in detail in §5.2.

Note (the relation between $d$ and $\int$—a boundary formula, not an inverse map) $\int_\gamma df = f(B)-f(A)$ does not mean that integration is a global inverse of the exterior derivative $d$. Integration is a functional that depends on the choice of curve $\gamma$; it is not an operation that reconstructs the entire original function $f$ from $df$. This formula is the fundamental theorem of calculus: "integrating an exact form along a curve leaves only the boundary, namely the endpoints." It is safest to think of it as the $0$-form version of the Stokes-type formulas we will see later.

§5.2 The "Mismatch" Revealed by a Closed Loop

5.2.1 What About a General $1$-Form?

What about a general $1$-form

? Here $P, Q, R$ are coefficients that vary from place to place (scalar fields)—the "general $1$-form" introduced in Chapter 3 §3.4.1.

Note (what "general" means) The $df$ treated in §5.1 had coefficients of the special combination $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}$—a triple derived from partial derivatives of a single function $f$. By contrast, the "general" $\omega$ here has $P, Q, R$ as three arbitrary scalar fields unrelated to one another. In other words, we call a $1$-form "general" when it cannot necessarily be written as $\omega = df$. The $y\,dx + x\,dy$ of §5.1.3 happened to be an exact form writable as $d(xy)$, but forms that are not of that kind are rather the norm.

$\omega$ need not represent a force; it can be any measuring device applied along a line.

For this $\omega$, there is no guarantee that the same telescoping sum as in §5.1 works. The most concise way to decide is to integrate on a closed loop—a curve whose start and end coincide. If $\int_\gamma \omega$ could be written only as the difference of endpoint values, then on a closed loop we would have to get $f(A)-f(A)=0$.

We write the line integral along a closed loop as $\oint_\gamma \omega$. In general:

If you drag an object once around against friction, the work is not zero; if you go once around a swirling current, you receive net work. This fact that "closing the loop still does not balance the books" is evidence of local structure such as circulation or vorticity.

Note (no physics background needed) Readers who have not yet treated friction or water flow in physics need not stop here. What matters now is not the specific physical phenomenon but only the sense that "if the integral on a closed loop is not zero, there is local structure like circulation in that field." The detailed physical correspondences will be touched on in later chapters.

5.2.2 Where Does the Local Information Live?

Then where does this "mismatch left after one circuit" come from? If we think of the closed loop only at a global scale all at once, we cannot tell which points inside contributed how much to the mismatch. Recall how we found $f'(x)$ in Chapter 1—we extracted a rate of change at a single point by taking the limit $\Delta x \to 0$ of $\Delta f / \Delta x$.

We use the same idea here. As we shrink the loop more and more, what happens to the value of the integral $\oint \omega$ around one circuit? If the leading term of $\oint \omega$ shrinks in proportion to the area enclosed by the loop, then the mismatch per unit area—how much the books fail to balance per unit area—should be determined as a value intrinsic to that point.

Conversely, if it does not scale that way (for example, if it scales with the perimeter of the loop), it cannot be fixed as a per-area quantity, and we cannot call it "a property at that point." So the question is:

When a tiny loop is traversed once, does the leading term of $\oint \omega$ scale with the enclosed area? If so, what is the proportionality constant?

In the next section we actually answer this question with a tiny rectangle placed in the $xy$ plane.

§5.3 Dismantling an Infinitesimal Loop — The Mismatch Is Proportional to Area

5.3.1 A Tiny Rectangle in the $xy$ Plane

Consider a tiny rectangle parallel to the $xy$ plane, with the point $(x, y, z)$ as its lower-left vertex. Let the width in the $x$ direction be $\Delta x$ and the width in the $y$ direction be $\Delta y$. Traverse the four edges of this rectangle once in the counterclockwise (right-handed) direction, and integrate $\omega = P\,dx + Q\,dy + R\,dz$.

Let us evaluate the integral on each edge using a first-order Taylor approximation (with $\Delta x, \Delta y$ sufficiently small).

Note (on Taylor expansion) There is no need to be intimidated by the words "Taylor expansion." What we use here is the first-order approximation of a multivariable function—the same thing as $\Delta f \approx f'(x)\,\Delta x$ in Chapter 1 §1.2.1. For example, the value of $Q(x+\Delta x, y)$ shifted by $\Delta x$ in the $x$ direction can be approximated as $Q + \frac{\partial Q}{\partial x}\,\Delta x$ using the partial derivative of $Q$ with respect to $x$. When $\Delta x$ is sufficiently small, terms of order $(\Delta x)^2$ and higher are overwhelmingly smaller than $\Delta x$ and may be ignored—that is what first-order approximation means.

$z$ is fixed, so $dz=0$ and the $R$ term contributes nothing.

Edge 1 (bottom edge, rightward): from $(x, y)$ to $(x+\Delta x, y)$. With $y$ fixed, $dy=0$. $P$ is approximately $P(x, y, z)$. Contribution: $P(x, y, z)\,\Delta x$.

Edge 2 (right edge, upward): from $(x+\Delta x, y)$ to $(x+\Delta x, y+\Delta y)$. $dx=0$. Evaluate $Q$ at the position shifted by $\Delta x$ in the $x$ direction:

Contribution: $\bigl(Q + \frac{\partial Q}{\partial x}\Delta x\bigr)\,\Delta y$.

Edge 3 (top edge, leftward): from $(x+\Delta x, y+\Delta y)$ to $(x, y+\Delta y)$. The direction is negative, so the sign flips. Evaluate $P$ at the position shifted by $\Delta y$ in the $y$ direction:

Contribution: $-\bigl(P + \frac{\partial P}{\partial y}\Delta y\bigr)\,\Delta x$.

Edge 4 (left edge, downward): from $(x, y+\Delta y)$ back to $(x, y)$. Contribution: $-Q(x, y, z)\,\Delta y$.

Summing the contributions from all four edges:

The omitted terms are higher order in $\Delta x$ and $\Delta y$.

The terms $P\Delta x$ and $Q\Delta y$ cancel beautifully, leaving only $(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})\,\Delta x\,\Delta y$ as the leading term.

5.3.2 The Decisive Fact

This result tells us just one thing:

The leading term of the mismatch left after one circuit is proportional to the enclosed area $\Delta x\,\Delta y$.

And the proportionality constant $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$ is precisely the mismatch per unit area at the point $(x,y,z)$—a local quantity representing the "strength of vorticity" at that point. More precisely, for the tiny rectangle $R_{\Delta x,\Delta y}$,

is the quantity extracted by this limit.

This is "differentiation" in the same spirit as $f'(x) = \lim_{\Delta x\to 0} \frac{\Delta f}{\Delta x}$ in Chapter 1. Only the denominator has changed—from "distance moved" to "area enclosed"—but the spirit of extracting a rate of change locally is unchanged.

Note (why only $\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}$ survives) $\frac{\partial P}{\partial x}$ and $\frac{\partial Q}{\partial y}$ do not appear. The change of $P$ in the $x$ direction ($\frac{\partial P}{\partial x}$) acts in the same direction on the bottom and top edges and cancels; the change of $Q$ in the $y$ direction ($\frac{\partial Q}{\partial y}$) likewise cancels on the right and left edges. What survives is only "the change of $P$ in the $y$ direction" and "the change of $Q$ in the $x$ direction"—the difference of partial derivatives in mutually orthogonal directions. This crosswise structure is the key to everything from the next section onward.

§5.4 The Birth of $d$ — A New Measuring Device for Mismatch per Unit Area

5.4.1 Revisiting the Wedge Product

Let us restate the $(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})\,\Delta x\,\Delta y$ obtained in §5.3 in the language of Chapter 2. In Chapter 2 §2.4.4 we defined the area-measuring device $dx \wedge dy$. This was a device that, when fed two vectors $\mathbf{v}_1, \mathbf{v}_2$, returns the signed area of the shadow they cast onto the $xy$ plane:

In particular, if we feed the displacement $\Delta x\,\hat{e}_x$ in the $x$ direction and the displacement $\Delta y\,\hat{e}_y$ in the $y$ direction, we get $(dx \wedge dy)(\Delta x\,\hat{e}_x,\; \Delta y\,\hat{e}_y) = \Delta x\,\Delta y$.

5.4.2 Definition of $d\omega$

Written in this language, the result of §5.3 says that for the two edges $\Delta x\,\hat{e}_x,\; \Delta y\,\hat{e}_y$ of an infinitesimal rectangle, some new $2$-form returned the leading term $(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})\,\Delta x\,\Delta y$. We call this $2$-form the exterior derivative of $\omega$, and write it $d\omega$:

$d\omega$ is a $2$-form—it eats two vectors and returns a scalar. Its value represents the mismatch per unit area of a closed loop over the infinitesimal parallelogram spanned by the two vectors fed to it.

Let us organize what is happening here:

• Input: $\omega = P\,dx + Q\,dy + R\,dz$ ($1$-form, a line measuring device that eats one vector)
• Output: $d\omega = (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})\,dx \wedge dy$ ($2$-form, a surface measuring device that eats two vectors)
• Operation: $d$ raises the degree by one, from 1 to 2

This is exactly the same structure as the $df$ we saw in §5.1. $d$ acts on a $0$-form $f$ and returns the $1$-form $df$; it acts on a $1$-form $\omega$ and returns the $2$-form $d\omega$. One of the great roles of $d$ is to raise the degree by one.

Checkpoint

- $df$ was the exterior derivative from $0$-form to $1$-form. $\int_\gamma df = f(B)-f(A)$ (depends only on the endpoints).

- For a general $\omega$, $\oint \omega \neq 0$ on a closed loop. To investigate this mismatch, we shrink the loop.

- On an infinitesimal rectangle in the $xy$ plane, $\oint \omega = (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})\,\Delta x\,\Delta y +$ higher-order terms. The leading term of the mismatch is proportional to area.

- We define the $2$-form that measures this proportionality coefficient as $d\omega := (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})\,dx \wedge dy$. $d$ is the operator that raises the degree.

§5.5 The Exterior Derivative of a General $1$-Form — Extension to Three Dimensions

5.5.1 The $yz$ Plane and the $zx$ Plane

In §5.3–§5.4 we considered loops only in the $xy$ plane. But in three-dimensional space there are three orientations of surface. On an infinitesimal rectangle in the $yz$ plane, a similar calculation gives $(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z})\,\Delta y\,\Delta z$; on the $zx$ plane we get $(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x})\,\Delta z\,\Delta x$. These correspond to $dy \wedge dz$ and $dz \wedge dx$, respectively.

The complete formula that follows from this intuition is:

Note ($2$-forms measure tilted parallelograms) This $d\omega$ is a $2$-form—a measuring device that eats two vectors. The two vectors fed to it need not be parallel to the $xy$ plane. If we feed the two edges $\mathbf{v}_1, \mathbf{v}_2$ of an infinitesimal parallelogram floating obliquely in three-dimensional space, it returns the mismatch per unit area of a closed loop over that surface. It is exactly the same thing we did in §5.3—the only difference is that the surface being measured need not be parallel to a coordinate plane. The mechanism for extracting the area (and orientation) of a parallelogram from two vectors was already verified in Chapter 2 §2.4, when we constructed $dy \wedge dz$ and the like as antisymmetric matrices.

5.5.2 Algebraic Derivation — Leibniz Rule and $d(dx)=0$

The formula above can of course be derived by doing separate loop calculations on each plane. But drawing a loop and adding four edges every time is tedious. What we do here is recast the exterior derivative introduced geometrically in §5.3 as computational algebraic rules, and confirm that these rules do indeed reproduce the exterior derivative. In other words, using the intuition from the $xy$-plane example, we move to a form that can be computed mechanically.

Note (Relation to the axiomatic definition) Many mathematics books take Leibniz's rule and $d(dx)=0$ as the definition of $d$. That definition is concise, but it is hard to see why those rules should hold. In this book we start from the dissection of an infinitesimal loop. What we are about to verify is that this geometric $d$ and the algebraic computational rules are consistent. The rules are for computational convenience; the meaning is in §5.3. Still, when one looks ahead to higher dimensions, this algebraic definition is indeed very concise.

First, for $\omega = P\,dx + Q\,dy + R\,dz$, we take $d$ to act linearly: $d\omega = d(P\,dx) + d(Q\,dy) + d(R\,dz)$. This is the same as considering loops around tilted parallelograms. The question is $d(P\,dx)$—how does $d$ act on the $1$-form $dx$ with coefficient $P$?

Here, the geometric $d$ obtained in §5.3–§5.4 has already taught us something. For $\omega = P\,dx$ ($Q=R=0$), if we carry out the infinitesimal loop calculation of §5.3 not only in the $xy$ plane but also in the $zx$ plane, $d(P\,dx)$ should have no component on the $yz$ face, and

(the loop on the $xy$ face gives $(0 - \frac{\partial P}{\partial y}) = -\frac{\partial P}{\partial y}$; the loop on the $zx$ face gives $(\frac{\partial P}{\partial z} - 0) = \frac{\partial P}{\partial z}$).

Now recall $dP = \frac{\partial P}{\partial x}\,dx + \frac{\partial P}{\partial y}\,dy + \frac{\partial P}{\partial z}\,dz$ (§5.1). Taking its wedge product with $dx$:

(the term with $\frac{\partial P}{\partial x}$ vanishes because $dx \wedge dx = 0$)

This agrees completely with the geometric calculation. That is, at least for a term of the form $P\,dx$:

So only the variation of the coefficient $P$ from place to place attaches to the measuring device in the $dx$ direction as a new area-measuring device.

Comparing the geometric result with Leibniz's rule $d(P\,dx) = dP \wedge dx + P\,d(dx)$, the extra term $P\,d(dx)$ must be zero. Since $P$ is an arbitrary function, this can hold in general only if $d(dx)=0$—that is, $d(dx) = 0$ is required.

Generalizing this observation, we arrive at the following graded Leibniz rule. $P$ is a $0$-form and $dx$ is a $1$-form; the behavior of $d$ on the “product” of a $0$-form and a $1$-form is:

$dP$ is known—from §5.1, $dP = \frac{\partial P}{\partial x}\,dx + \frac{\partial P}{\partial y}\,dy + \frac{\partial P}{\partial z}\,dz$. The question is the value of $d(dx)$.

Here, the exterior derivative of the coordinate function $x$ is $dx$ ($d(x) = dx$). The $dx$ of Cartesian coordinates is a reference measuring device whose coefficients do not change from place to place. So even if we send $dx$ itself once around an infinitesimal loop, no mismatch per unit area appears. The same holds for $dy$ and $dz$. Therefore, for the coordinate basis, we adopt the following computational rule:

Then:

Using the antisymmetry of $\wedge$ ($dx \wedge dx = 0$, $dy \wedge dx = -dx \wedge dy$):

Similarly:

Adding the three and collecting in the order of the basis $dy \wedge dz,\; dz \wedge dx,\; dx \wedge dy$ (this cyclic order is foreshadowing for consistency with the right-handed system in the next chapter):

This includes the $xy$ component $(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})$ derived geometrically in §5.3, and the $yz$ and $zx$ components follow the same cross pattern.

5.5.3 The Pattern of the Coefficients

Notice how the coefficients are arranged. $(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z},\; \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x},\; \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})$ are obtained by taking partial derivatives of $(P, Q, R)$ with respect to axes other than their own, crossing them, and taking differences. There is a cyclic symmetry:

Note (A guidepost for experienced readers) Readers who already know vector analysis will find these three coefficients familiar. In this book we do not rely on their names; we first build the operation $d$ itself. The correspondence will be organized in later chapters.

Checkpoint

- The exterior derivative $d\omega$ of a general $1$-form $\omega$ is a $2$-form whose coefficients are the cross differences of partial derivatives $(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z},\; \frac{\partial P}{\partial z}-\frac{\partial R}{\partial x},\; \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})$.

- The computational rules are threefold: (1) linearity, (2) Leibniz rule $d(P\,dx) = dP \wedge dx + P\,d(dx)$, (3) $d(dx)=d(dy)=d(dz)=0$.

- The result is fully consistent with the infinitesimal loop calculation of §5.3.

§5.6 Accumulate It, and Only the Boundary Remains — Stokes' Theorem

5.6.1 Tiling a Surface

In §5.3 we computed $\oint \omega$ around a single infinitesimal rectangle and saw that its leading term is the product of $d\omega$ and area. What happens if we tile an entire surface with these infinitesimal rectangles and add everything up?

Divide a surface $S$ into small coordinate patches and cut each into infinitesimal rectangles on its parameter plane (this is exactly the idea of a surface element from Chapter 3 §3.2). For each infinitesimal rectangle, the integral of $\omega$ around the boundary in the orientation matching the surface has, as its leading term, the value obtained by feeding the surface element to $d\omega$.

Here we consider the case where the surface can be smoothly parameterized and a natural orientation is induced on the boundary as well. Reversing the orientation of the surface simultaneously reverses the direction in which the boundary is traced. Only after fixing this “pairing of surface orientation and boundary orientation” do the signs on the two sides agree.

5.6.2 Interior Edges Cancel

Here is the decisive geometric observation. Focus on the edge shared by two adjacent rectangles. For the rectangle on the left, that edge is traced “upward”; for the rectangle on the right, it is traced “downward.” Because the paths run in opposite directions, the line integral of $\omega$ cancels completely on this edge.

This cancellation occurs on every shared edge in the interior of the surface $S$. No matter how finely we partition and add, the contributions from interior edges vanish in plus-minus pairs.

5.6.3 Only the Boundary Survives

Then which edges survive without cancellation—edges for which no adjacent rectangle exists, that is, edges along the boundary $\partial S$ of the surface $S$.

In other words:

The left-hand side is “the line integral of $\omega$ along the one-dimensional closed curve that is the boundary of the surface”; the right-hand side is “the sum of mismatch densities from the countless infinitesimal loops tiled in the interior of the surface.” In the limit of a finer and finer partition, only the cancellation of interior edges remains, and the two sides agree.

This is the Kelvin–Stokes theorem (also called simply Stokes' theorem).

Writing out the components for $\omega = P\,dx + Q\,dy + R\,dz$ explicitly:

In the matrix language we have used since Chapter 2, the $2$-form on the right-hand side is represented by the antisymmetric matrix $\mathbf{J}^T - \mathbf{J}$, where $\mathbf{J}$ is the Jacobian matrix of the coefficient vector $(P,Q,R)$. If $\mathbf{v}_1, \mathbf{v}_2$ are the two edges of a surface element, then $(d\omega)(\mathbf{v}_1, \mathbf{v}_2) = \mathbf{v}_1^T(\mathbf{J}^T - \mathbf{J})\mathbf{v}_2$. All components are written out in Appendix B.3; consult it as needed.

The structure of this theorem is remarkably simple. Integral measured on the global boundary $=$ accumulation of local mismatch ($d\omega$) in the interior. $d$ is the device that translates information about $\omega$ on the boundary $\partial S$ into information about $d\omega$ in the interior $S$.

Note (Assumptions used in localization) The tiling explanation here assumes a sufficiently smooth surface and field that can be divided into sufficiently fine coordinate patches. When there are cusps on the boundary, self-intersections, singular points, and the like, patch decomposition and orientation must be handled separately. In this book we treat the smooth cases ordinarily used in physics.

Note (Correspondence with Chapter 3) When we defined surface integrals in Chapter 3 §3.2.1, we fed $dx \wedge dy$ to the surface element $(\mathbf{r}_u\Delta u,\;\mathbf{r}_v\Delta v)$. Here too, we divide the surface into small surface elements, act with the measuring device on each element, and organize the contributions that cancel in the interior.

Checkpoint

- If we tile a surface $S$ with infinitesimal loops, interior edges cancel and only the contribution from the boundary $\partial S$ remains.

- $\oint_{\partial S} \omega = \iint_S d\omega$. A global boundary integral equals an integral of the local exterior derivative.

- As with surface integrals in Chapter 3, we divide the surface into small patches, act with the measuring device on each patch, and aggregate.

§5.7 The Same Thing One Degree Higher — Exterior Derivative of a $2$-Form and Divergence

5.7.1 A $2$-Form Is a Measuring Device for Surfaces

In Chapter 3 §3.4.2 we wrote a general $2$-form in the form

This is a measuring device for flux through a surface. As we saw in Chapter 2 §2.4.5, the three basis $2$-forms each measure the area of the projection onto the $yz$, $zx$, and $xy$ planes, respectively.

5.7.2 Measuring on the Faces of a Tiny Rectangular Box

In the same spirit as §5.3, we now integrate $\eta$ over all six faces of a tiny rectangular box in space, $[x, x+\Delta x] \times [y, y+\Delta y] \times [z, z+\Delta z]$. We align the orientations outward.

For the term $C\,dx \wedge dy$, consider only the bottom and top faces perpendicular to $z$:

• Bottom face ($z$, outward direction is $-z$): evaluate $C(x,y,z)$ with the orientation of $-dx \wedge dy$ → approximately $-C(x,y,z)\,\Delta x\,\Delta y$
• Top face ($z+\Delta z$, outward direction is $+z$): $C(x,y,z+\Delta z) \approx C + \frac{\partial C}{\partial z}\Delta z$ → approximately $(C + \frac{\partial C}{\partial z}\Delta z)\,\Delta x\,\Delta y$

Summing, the $C\Delta x\Delta y$ terms cancel and $\frac{\partial C}{\partial z}\,\Delta x\,\Delta y\,\Delta z$ remains. The term $A\,dy \wedge dz$ contributes $\frac{\partial A}{\partial x}\,\Delta x\,\Delta y\,\Delta z$ from the faces perpendicular to $x$, and the term $B\,dz \wedge dx$ contributes $\frac{\partial B}{\partial y}\,\Delta x\,\Delta y\,\Delta z$ from the faces perpendicular to $y$.

The total over all six faces is:

Here too the structure is the same—the leading term of the mismatch measured on the surface is proportional to the enclosed volume. The proportionality constant $\frac{\partial A}{\partial x} + \frac{\partial B}{\partial y} + \frac{\partial C}{\partial z}$ represents the "outflow per unit volume."

5.7.3 Definition of $d\eta$ and Algebraic Derivation

As a $3$-form that eats the three edges $\Delta x\,\hat{e}_x,\; \Delta y\,\hat{e}_y,\; \Delta z\,\hat{e}_z$ of the box:

This formula can also be derived from the algebraic rules of §5.5.2. First, as we did in §5.5.2, let us confirm for the term $A\,dy \wedge dz$ that the geometric result is consistent with the Leibniz rule.

According to the geometric calculation in §5.7.2, the contribution of $A\,dy \wedge dz$ was $\frac{\partial A}{\partial x}\,\Delta x\,\Delta y\,\Delta z$. In the language of $dx \wedge dy \wedge dz$, this is $\frac{\partial A}{\partial x}\,dx \wedge dy \wedge dz$.

On the other hand, applying the Leibniz rule to $d(A\,dy \wedge dz)$ gives:

Here, as in §5.5.2, using $d(dy)=d(dz)=0$:

so the second term vanishes and only the first remains:

By antisymmetry of $\wedge$ ($dy \wedge dy = 0$, $dz \wedge dz = 0$), the terms with $\frac{\partial A}{\partial y}$ and $\frac{\partial A}{\partial z}$ vanish, leaving only $\frac{\partial A}{\partial x}\,dx \wedge dy \wedge dz$. This agrees completely with the geometric calculation in §5.7.2.

The terms $B\,dz \wedge dx$ and $C\,dx \wedge dy$ are similar; adding the three yields:

$d$ has accomplished the promotion $2$-form $\to$ $3$-form.

5.7.4 Gauss' Theorem

By the same tiling argument as in §5.6—this time filling the solid $V$ with tiny rectangular boxes—internal faces cancel and only the outer surface $\partial V$ remains:

This is Gauss' theorem. It has exactly the same form as Stokes' theorem, only with the dimension shifted by one. Placed side by side, the parallel is obvious: $\omega$ ($1$-form) $\to$ $d\omega$ ($2$-form) $\to$ surface $S$, $\eta$ ($2$-form) $\to$ $d\eta$ ($3$-form) $\to$ solid $V$.

Writing out the components explicitly for $\eta = A\,dy \wedge dz + B\,dz \wedge dx + C\,dx \wedge dy$:

Note (a guide for readers already familiar with vector calculus) Readers who already know vector analysis may recognize $\frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z}$. In this book we first obtain this quantity as the exterior derivative of a $2$-form, and we keep that order of presentation. The correspondence with names will be organized in a later chapter.

Checkpoint so far

- The exterior derivative $d\eta$ of a $2$-form $\eta$ is a $3$-form whose coefficient is $\frac{\partial A}{\partial x} + \frac{\partial B}{\partial y} + \frac{\partial C}{\partial z}$.

- The derivation can be done mechanically with the same algebraic rules as §5.5 (Leibniz rule $+$ $d(dx)=0$).

- $\iint_{\partial V} \eta = \iiint_V d\eta$. Stokes ($1$-form $\to$ surface) and Gauss ($2$-form $\to$ solid) differ only in degree; the structure is the same.

§5.8 $d^2 = 0$ — The Mismatch of a Mismatch Leaves Nothing Behind

Before unifying Stokes' theorem and Gauss' theorem in the next section, let us confirm one more basic structure of the exterior derivative. $d$ raises the degree by one, but applying it twice in succession leaves no new mismatch. This fact is also connected to the geometric structure we will see later: "the boundary of a boundary cancels as an oriented sum."

5.8.1 $d(df) = 0$

In §5.1 we defined $df = \frac{\partial f}{\partial x}\,dx + \frac{\partial f}{\partial y}\,dy + \frac{\partial f}{\partial z}\,dz$. Let us apply the exterior derivative once more. Using the formula from §5.5 with $P = \frac{\partial f}{\partial x},\; Q = \frac{\partial f}{\partial y},\; R = \frac{\partial f}{\partial z}$:

By symmetry of mixed partial derivatives (if $f$ is $C^2$ (twice continuously differentiable), then $\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}$, etc.; see Chapter 1 §1.2), every coefficient vanishes:

Note ($d^2=0$ in the axiomatic viewpoint and the $C^2$ condition) When we compute concretely in coordinates, we see that $d^2=0$ is inevitable from symmetry of mixed partial derivatives and antisymmetry of the wedge product. From the axiomatic standpoint of differential forms, one may instead read things the other way around: requiring $d^2=0$ (as part of the definition) means we are implicitly working with a class of functions for which symmetry of mixed partial derivatives holds ($C^2$ functions). In this book we first understand it through concrete calculation, as the mechanism that prevents contradictions in the hierarchy of physical laws.

5.8.2 $d(d\omega) = 0$

Apply the formula of §5.7 to the $d\omega$ of §5.5. With $A = \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z},\; B = \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x},\; C = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$:

Expanding and rearranging the expression in parentheses:

By symmetry of mixed partial derivatives ($\frac{\partial^2 R}{\partial y \partial x}=\frac{\partial^2 R}{\partial x \partial y}$, etc.), the six terms cancel in three pairs and the sum is zero:

Checkpoint so far

- $d^2 = 0$: applying the exterior derivative twice always gives zero. Symmetry of mixed partial derivatives and antisymmetry of the wedge produce the cancellation.

- Several identities familiar to readers who already know vector calculus—"it vanishes when you differentiate twice"—will be organized in a later chapter as other expressions of this $d^2=0$.

§5.9 Unifying the Exterior Derivative — One Formula, One Rule

5.9.1 Stokes and Gauss Were the Same Formula

Place Stokes' theorem $\oint_{\partial S}\omega = \iint_S d\omega$ from §5.6 and Gauss' theorem $\iint_{\partial V}\eta = \iiint_V d\eta$ from §5.7 side by side. Only the number of integral signs differs—$\oint$, $\iint$, $\iiint$—yet the structure is identical. If $M$ is a curve, we write $\int$ (1-dimensional); if a surface, $\iint$ (2-dimensional); if a solid, $\iiint$ (3-dimensional)—the number of integral signs is determined only by the dimension of $M$.

So, regardless of dimension, we write uniformly:

Here $\omega$ is a $k$-form, $M$ is a $(k+1)$-dimensional region, and $\partial M$ is its $k$-dimensional boundary. For $k=0$, with the oriented boundary $\partial M=\{B\}-\{A\}$, this is the fundamental theorem of calculus: $\int_{\partial M} f=f(B)-f(A)=\int_M df$; for $k=1$ it is Stokes' theorem; for $k=2$ it is Gauss' theorem. All fit into this single line.

The power of this unified form stands out even more in combination with $d^2=0$ from §5.8. Replacing $\omega$ by $d\omega$ in $\int_{\partial M} \omega = \int_M d\omega$:

That is, "the boundary of a boundary is zero as an oriented sum" ($\partial^2 M = 0$) and $d^2=0$ correspond to each other. This does not mean that the second boundary is always empty as a set. For example, if we subdivide the boundary of a polygon further into boundary pieces, vertices appear—but counted with orientation, contributions from adjacent edges cancel. $d^2=0$ is the algebraic portrait of this topological cancellation.

Note (three dimensions suffice) Because the stage of this book is three-dimensional space, the dimension of $M$ is 1, 2, or 3, and $\omega$ is a $0$-form, $1$-form, or $2$-form. We do not enter $k=3$ ($M$ four-dimensional). Still, it does no harm to keep in the back of your mind that this formula holds regardless of $k$.

5.9.2 Exterior Derivative of a General $k$-Form

Likewise, the action of $d$ collapses into one pattern regardless of degree. When an arbitrary $k$-form $\omega$ is written as a sum of coefficients $a_{i_1\cdots i_k}$ and basis elements $dx_{i_1}\wedge\cdots\wedge dx_{i_k}$:

For this book we need only the three cases $k=0,1,2$, written out concretely in §5.1 ($df$), §5.5 ($d\omega$), and §5.7 ($d\eta$). The general form above is only confirmation that "at every degree, take the total differential of the coefficients and attach with the wedge"—a single principle.

5.9.3 Summary of Computational Rules

All computations of the exterior derivative $d$ built in this chapter are summarized in the following four rules:

1. Action on a $0$-form: $df = \frac{\partial f}{\partial x}\,dx + \frac{\partial f}{\partial y}\,dy + \frac{\partial f}{\partial z}\,dz$ (the definition since Chapter 1 §1.2)
2. Linearity: $d(\omega_1 + \omega_2) = d\omega_1 + d\omega_2$
3. Graded Leibniz rule: $d(f\,\omega) = df \wedge \omega + f\,d\omega$
4. Exterior derivative of coordinate bases is zero: $d(dx) = d(dy) = d(dz) = 0$

Rule 4 reflects the geometric fact that the coordinate bases $dx, dy, dz$ themselves have no coefficients that vary with position, so measuring them on a tiny loop produces no local mismatch. It is also consistent with $d^2=0$ in §5.8.

With only these four rules, the exterior derivative of any form from $0$-form to $3$-form can be computed mechanically. In fact, every calculation treated in this chapter is nothing but a combination of these rules.

Note (commutativity with pullback—payoff of a Chapter 4 foreshadowing) As foreshadowed in Chapter 4, between pullback $\Phi^*$ and exterior derivative $d$ there holds the commutation relation

$$\Phi^*(d\omega) = d(\Phi^*\omega)$$

"differentiate then pull back" and "pull back then differentiate" give the same result. This property cannot be derived directly from the four rules above, but if we recall that $d$ is the operation that "measures local mismatch" and $\Phi^*$ is the operation that "relabels coordinates," it is natural—the structure of local mismatch does not depend on how coordinates are chosen. This fact will be an important footing when we consider differential forms on general manifolds in Chapter 11.

Checkpoint so far

- A single line $\int_{\partial M} \omega = \int_M d\omega$ unifies the fundamental theorem of calculus, Stokes' theorem, and Gauss' theorem.

- Computation of $d$ is summarized in four rules ($df$, linearity, Leibniz rule, $d(dx)=0$).

- $d^2=0$ and $\partial^2 M=0$ echo each other as algebraic truth and geometric truth.

§5.10 From Integrals to Differential Equations — Localizing Physical Laws

5.10.1 Turning Integral Laws into Differential Laws

Let us return to the question posed at the opening of this chapter.

How do we extract a local law from an integrated quantity?

The answer lies in the two tools we have developed—Stokes' theorem and the exterior derivative $d$.

Many fundamental laws of physics are discovered first in integral form. They take the shape: a quantity measured on the boundary of a region equals the total of sources inside:

The left-hand side is the integral measured on the boundary (a global observable); the right-hand side is the integral of sources in the interior (a global total). $\omega$ and $\eta$ are differential forms of the appropriate degrees.

Apply Stokes' theorem to the left-hand side. Since $\int_{\partial M} \omega = \int_M d\omega$:

Rearranging:

This is the decisive moment. If this equality holds for every smooth region $M$, including sufficiently small ones, the integrand form must vanish identically at each point. For if there were a point where $d\omega - \eta \neq 0$, the integral over a tiny region around that point would not be zero—a contradiction. Here too we are considering the case where the fields are sufficiently smooth and contain no singular concentrated sources.

Therefore:

This is the local differential law extracted from the integral law. $d$ has functioned precisely as the operator that converts an integral equation into a differential equation.

In other words, the exterior derivative $d$ is not merely a symbol that raises the degree of a form. It is the operator that translates a law observed on the boundary into a law that holds at each point in the interior.

5.10.2 Examples from Physics

A typical instance of this pattern is the fundamental laws of electromagnetism.

Note (for readers who have not yet studied electromagnetism) In this section we are not after the detailed content of electromagnetism, but after the mechanism by which an integral law becomes a local one. Think of $E,D,B,H,J,\rho$ not so much as names of physical quantities as symbols standing for "quantities measured along a line," "quantities measured through a surface," and "quantities measured over a volume."

Note (for readers who learned electromagnetism through vector calculus) Here we choose the degree of each form according to the dimension of the object over which we integrate, rather than treating the fields as ordinary three-component vectors. Because $E$ and $H$ are integrated along lines, they are $1$-forms; because $D,B,J$ are integrated through surfaces, they are $2$-forms; because $\rho$ is integrated over volume, it is a $3$-form. The correspondence with the usual vector-calculus notation will be organized after we introduce the Hodge star.

In electromagnetism, the object of integration depends on the kind of quantity: some are measured along lines, some through surfaces, some over volume. Schematically, we can read the situation as follows.

For example, Gauss's law for charge takes the form

The left-hand side is the total flux of $D$ passing outward through the closed surface $\partial V$; the right-hand side is the total charge contained in the interior $V$. By Stokes' theorem,

so

If this holds for every region $V$, then

Similarly, the law that the net magnetic flux through a closed surface is zero can be written

By Stokes' theorem,

and if this holds for every $V$, then

Laws that include time variation have the same type. For a surface $S$,

implies

and

implies

Here $d$ is the exterior derivative in the spatial directions, and time variation is represented by $\partial/\partial t$.

Thus the fundamental equations of electromagnetism appear, in the language of this chapter, as expressions of the same type:

What matters here is not to memorize these as new formulas of electromagnetism. The single point is that an integral law measured on the boundary passes, via the exterior derivative $d$, to a local law in the interior.

On the other hand, relating $E$ to $D$, or $B$ to $H$, requires a convention for how space and the medium are measured. Likewise, matching quantities measured along lines to quantities measured through surfaces requires additional rules for how length, area, and volume are measured. The Hodge star, which we introduce in the next chapter, is the tool that supplies this correspondence.

A detailed development including component formulas is deferred to Appendix C, "Integral Forms of Electromagnetism and Differential Forms."

Checkpoint so far

- Physical laws are often given in the form $\int_{\partial M} \omega = \int_M \eta$.

- By Stokes' theorem, rewrite $\int_{\partial M} \omega = \int_M d\omega$; since the equality holds for every $M$, we obtain $d\omega = \eta$ (the local law).

- This localization, however, is an argument in regions where the fields are sufficiently smooth and contain no singular concentrated sources.

- $d$ is the operator that converts integral laws into differential laws.

§5.11 Outlook Toward Part II — Foreshadowing the Hodge Star

In this chapter we have acquired the powerful operator called the exterior derivative $d$. $d$ maps:

• $0$-form $\to$ $1$-form: $df = \frac{\partial f}{\partial x}\,dx + \frac{\partial f}{\partial y}\,dy + \frac{\partial f}{\partial z}\,dz$
• $1$-form $\to$ $2$-form: $d\omega = (\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z})\,dy\wedge dz + (\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x})\,dz\wedge dx + (\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})\,dx\wedge dy$
• $2$-form $\to$ $3$-form: $d\eta = (\frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z})\,dx\wedge dy\wedge dz$

Each time the degree rises by one step, the coefficients rearrange in a different pattern. And $d^2=0$ guarantees that no contradiction can slip in among these levels.

Yet one asymmetry stands out. A $0$-form and a $3$-form each have 1 independent component; a $1$-form and a $2$-form each have 3—symmetry in the pattern $1, 3, 3, 1$ (as we saw in Chapter 2 §2.5.9–§2.5.10). This symmetry is no accident. In three-dimensional space with Cartesian coordinates, once we add rules for measuring length, area, and volume, a "dictionary" appears that links forms of the same information content.

That dictionary is the Hodge star $\ast$, which we introduce in the next chapter. But $\ast$ is not a mere relabeling of symbols. It depends on which direction corresponds to which face, how area and volume are measured, and how orientation is fixed. For that reason we do not yet use concrete correspondence formulas in this chapter.

What we have built so far is, after all, only $d$. $d$ raises the degree by one, extracts local mismatch, and converts integral laws into local laws. Yet $d$ alone cannot yet reconstruct the familiar operations of vector analysis. To match $1$-forms with $2$-forms, we need the additional rules for how length, area, and volume are measured in space—the metric.

In the next chapter we define those rules of "measurement" as the Hodge star $\ast$. To the $d$ constructed in Chapter 5 we add $\ast$ in Chapter 6, and use the two together to rebuild the familiar operations of vector analysis in later chapters. From here we move into the core of Part II—unmasking div, grad, and curl.

Checkpoint so far — Chapter 5

- The exterior derivative $d$ is the operator that maps a $k$-form to a $(k+1)$-form: $0$-form $\to$ $1$-form ($df$), $1$-form $\to$ $2$-form, $2$-form $\to$ $3$-form.

- There are four computational rules: the definition of $df$, linearity, the Leibniz rule $d(f\omega) = df\wedge\omega + f\,d\omega$, and $d(dx)=d(dy)=d(dz)=0$.

- The geometric origin is §5.3's dissection of a tiny loop: the mismatch around a closed loop scales with area, and the proportionality constant is $d\omega$.

- Stokes' theorem $\int_{\partial M} \omega = \int_M d\omega$ is the universal bridge linking boundary and interior.

- $d^2 = 0$ comes from the symmetry of mixed partial derivatives and the antisymmetry of the wedge product; geometrically it corresponds to "the boundary of a boundary is zero as an oriented sum."

- $d$ is the operator that converts integral laws into local differential laws—an indispensable role in the formulation of physics.

- In the next chapter we add rules for measuring length, area, and volume, and organize the correspondence between $d$ and the operations of vector analysis through the Hodge star $\ast$.

Appendix B: Matrix Representation of the Exterior Derivative

The main text of this chapter proceeded with the basis $dx, dy, dz$ and the algebra of the wedge product. Here we rewrite the exterior derivative in the language of matrix representations, following the book's convention since Chapter 2 — arranging every component without exception into matrices.

B.1 $0$-form: the $1 \times 3$ row vector of $df$

For $f = f(x,y,z)$, as defined in Chapter 1 §1.2:

As a row vector ($1 \times 3$ matrix):

B.2 $1$-form: the Jacobian matrix $\mathbf{J}$ of the coefficients

For $\omega = P\,dx + Q\,dy + R\,dz$, the $3 \times 3$ matrix that arranges all partial derivatives of the coefficients $(P,Q,R)$ is:

This is the Jacobian matrix of the vector field obtained by stacking $(P,Q,R)$ as a column. The reason §5.1 said that "a mere matrix of partial derivatives is not enough" is precisely that this $\mathbf{J}$ is not antisymmetric.

B.3 $d\omega = \mathbf{J}^T - \mathbf{J}$

The result of §5.5 is:

Following the convention of Chapter 2 §2.4.4, we seek the antisymmetric matrix $\mathbf{M}$ such that $(d\omega)(\mathbf{v}_1, \mathbf{v}_2) = \mathbf{v}_1^T \mathbf{M} \mathbf{v}_2$ for arbitrary column vectors $\mathbf{v}_1, \mathbf{v}_2$.

Writing out the entries of $\mathbf{J}^T - \mathbf{J}$:

With the Chapter 2 convention for representing a $2$-form by an antisymmetric matrix, this gives:

Thus, the matrix representation of the exterior derivative $d\omega$ is the antisymmetric matrix obtained by subtracting the Jacobian matrix of the coefficients from its transpose.

B.4 $2$-form: $d\eta$ and the trace of the Jacobian matrix

For $\eta = A\,dy \wedge dz + B\,dz \wedge dx + C\,dx \wedge dy$, let the Jacobian matrix of the coefficients $(A,B,C)$ be:

Then the coefficient in the result of §5.7, $d\eta = (\frac{\partial A}{\partial x} + \frac{\partial B}{\partial y} + \frac{\partial C}{\partial z})\,dx \wedge dy \wedge dz$, agrees with the trace (sum of diagonal entries) of $\mathbf{J}_\eta$:

B.4.1 The 27 components of $d\eta$ — expanded into three matrices

The antisymmetric components of $\eta$ are $\eta_{yz}=A,\; \eta_{zx}=B,\; \eta_{xy}=C$ (the others are sign reversals or $0$). To see where all entries go, first arrange the raw derivative components $\partial_a\eta_{bc}$, where $\partial_x=\frac{\partial}{\partial x}$, $\partial_y=\frac{\partial}{\partial y}$, $\partial_z=\frac{\partial}{\partial z}$, and $a,b,c\in\{x,y,z\}$, so there are $3^3=27$ entries before antisymmetrization. Arranging them into three $3 \times 3$ matrices with $a$ fixed gives:

When these $27$ components are contracted with the corresponding basis wedge products of $1$-forms (for example, if $a=x,\;b=y,\;c=z$, then $dx \wedge dy \wedge dz$), terms with repeated indices (diagonal entries, $b=c$, and so on) vanish because $dx \wedge dx = 0$, and only the $6$ terms with all of $a,b,c$ distinct (permutations of $3!$) survive. Each is summed with its sign. The factor convention is the same as in Chapter 2 §2.4.4: the matrix stores each antisymmetric pair twice, while the form stores it once. With that convention, the six surviving signed terms combine to

In other words, of the $27$ components of $d\eta$, $6$ survive and pair up and combine into the trace of $\mathbf{J}_\eta$.

A $3$-form has only the single basis element $dx \wedge dy \wedge dz$, so in this representation it reduces to a scalar coefficient.

B.5 $d^2 f = 0$ and the Hessian

The Hessian of a $0$-form $f$:

arranges the second partial derivatives of $f$. If $f$ is $C^2$, then $\mathbf{H}_f$ is a symmetric matrix ($\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}$, and so on).

$d(df) = 0$ is the §5.5 formula for $d\omega$ with $(P,Q,R) = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z})$. We have $\mathbf{J} = \mathbf{H}_f$, and because $\mathbf{H}_f$ is symmetric, $\mathbf{J}^T - \mathbf{J} = 0$; hence $d(df) = 0$. In the language of matrices, this is rephrased as "the antisymmetric part of the Hessian is zero."

Appendix C: Integral Forms of Electromagnetism and Differential Forms

This appendix verifies, in component form, the passage from the integral forms of the four fundamental equations of electromagnetism to their local forms. What we use here are the exterior derivative $d$ and the degree of a differential form. The metric of space does not appear in this localization itself.

C.1 Degrees of physical quantities

Quantities measured along a line are placed as $1$-forms, quantities measured through a surface as $2$-forms, and quantities measured over a volume as $3$-forms. In components:

To keep the notation simple, we write both the coefficient representing charge density and the $3$-form obtained by multiplying it by the volume form with the same symbol $\rho$.

$E$ and $H$ are $1$-forms measured along a line; $D$, $B$, and $J$ are $2$-forms measured through a surface; $\rho$ is a $3$-form measured over a volume.

C.2 Gauss's law for charge

The integral form is:

Applying the generalized Stokes' theorem to the left-hand side:

Therefore, if

holds for every region $V$, the local form is:

Expanding in components:

Hence:

C.3 The law of magnetic flux

The integral form stating that the net magnetic flux through a closed surface is zero is:

By the generalized Stokes' theorem:

If this holds for every region $V$, the local form is:

Expanding in components:

Therefore:

C.4 Faraday's law

The integral form is:

Applying the generalized Stokes' theorem to the left-hand side:

If this holds for every surface $S$, the local form is:

Expanding in components:

On the other hand,

so the components are:

C.5 Ampère–Maxwell's law

The integral form is:

Applying the generalized Stokes' theorem to the left-hand side:

If this holds for every surface $S$, the local form is:

Expanding in components:

Also,

so the components are:

C.6 Where a metric is needed

The four local forms above can be written using only the exterior derivative $d$ and the degree of differential forms. At this stage, no metric for measuring lengths or angles is used.

A metric becomes necessary when relating $E$ to $D$ and $B$ to $H$. It is also needed when pairing $1$-forms with $2$-forms so that the usual three-component fields can be treated as objects of the same kind.

The Hodge star, introduced in the next chapter, is the tool that provides this correspondence.