The Exterior Derivative

The exterior derivative $d$ generalizes the gradient, curl, and divergence into a single unified operator. It maps $p$-forms to $(p+1)$-forms.

For a 1-form $\omega = P\,dx + Q\,dy$ on $\mathbb{R}^2$, the exterior derivative is a 2-form:

$d\omega = \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dx \wedge dy$

The coefficient $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$ is the curl of the vector field $(P, Q)$.

Closed and Exact Forms

A form $\omega$ is:

• Closed if $d\omega = 0$ — the curl vanishes
• Exact if $\omega = df$ for some function $f$ — it is the differential of something

Stokes's theorem says: $\int_{\partial M} \omega = \int_M d\omega$.

Examples

Your Task

Implement exterior_d(P, Q) that computes the exterior derivative of the 1-form $\omega = P\,dx + Q\,dy$. Return a function of $(x, y)$ giving the coefficient of $dx \wedge dy$.