Exterior derivative

See Wikipedia
See differential forms.
It is an antiderivation of degree 1 on the exterior algebra.

1. For a 0-form $df$ is the differential of $f$
2. $d(df)=0$ for 0-forms (for the other differential forms can be deduced)
3. $d(\alpha \wedge \beta )=d\alpha \wedge \beta +(-1{)}^p(\alpha \wedge d\beta )$ where $\alpha$ is a $p$-form.

Interpretation

See also: visualization of k-forms.

Exterior derivative must be called the negative accumulation meter or production meter.

Case 1: 0-Forms

For a 0-form $f$ (a scalar function), $f$ associate values to points, and $df$ represents the net production of $f$ along a direction:
Example (1D water pipe with a source):
Consider a pipe ($\mathbb{R}$) with water flowing rightward and a point source at $x=3$ injecting 4 units of water per second.

• Let $f(x)$ be the total flow passing point $x$.
  • For $x<3$, $f(x)=0$ (no water yet).
  • For $x\ge 3$, $f(x)=4$ (water added by the source).
• The exterior derivative $df$ is a 1-form encoding the source:$$df=4\delta (x-3)dx,$$

where $\delta (x-3)$ is a Dirac-like concentration at $x=3$.

• Interpretation:
  • $df=0$ where no sources exist (since $f$ is constant).
  • At $x=3$, $df$ spikes to reflect the production of 4 units.
• Integration:

confirming the source’s contribution.

General Intuition:

• $df(X)\approx f(P_2)-f(P_1)$ measures the net change of $f$ along $X=\overrightarrow{P_1{P}_2}$, for $P_1,P_2$ infinitesimally close.
• If $df=0$, $f$ is conserved (no production/loss) along that direction.

Case 2: 1-forms

In the case of a 1-form $\omega$, which associate values to line elements (vectors), exterior derivative tell us how of "something" is being produced in a bivector $u\wedge v$ (a "2-direction"). If we restrict $\omega$ to this area element (with a kind of pullback), the 1-form can be understood like a kind of flow (a 1-form in a 2-dimensional space can be seen as line families at every point, which can be joined together if the 1-form is closed). In this context, $d\omega$ measures how much of this flow is being produced in $u\wedge v$. This is the idea of the infinitesimal Stokes' theorem

In a sense, the value of $d\omega$ in $u\wedge v$ is like measuring how the 1-form varies "along the bivector" $u\wedge v$.

Case 3: 2-forms and more

In the case of 2-forms, for example in ${\mathbb{R}}^3$, these can be visualized as "packages of 1-dimensional fibers" in a neighborhood of each point (in ${\mathbb{R}}^n$, they look like packages of $(n-2)$-dimensional fibers). The value a 2-form assigns to each bivector represents the density of intersections between those fibers and the bivector.

If we apply the exterior derivative of the 2-form to a volume element (a 3-vector), we get something completely analogous to what was described above: pulling back to that "infinitesimally small 3-dimensional space," the 2-form still appears as a kind of one-dimensional flow, and its differential tells us how much is being generated inside (what goes out minus what came in).

If instead of working in ${\mathbb{R}}^3$, we were in ${\mathbb{R}}^n$, the pullback of the 2-form to the 3-vector still appears as a one-dimensional flow—although globally in ${\mathbb{R}}^n$, it may not be.

Why $d^2=0$?

From here it shouldn't be difficult understand why

For 0-forms it is easy: if $\omega =df$, the 2-form $d\omega$ is computed evaluating $\omega$ in sides of the parallelogram $u\wedge v$, which in turn is evaluated in the "vertices". But the latter appear twice in the final computation, but with different sign.
For a 1-form is the same, but a bit more difficult to visualize. In this case $d(d\omega )$ is evaluated in a parallelepiped, and the computation rests, finally, at evaluation on the edges, which appear twice with opposite sign.

The interpretation of $d\omega$ as production (negative accumulation) of a flow in a $k$-vector makes Stokes' theorem trivial. By the way, this interpretation is in some sense given in @needham2021visual page 409.