Exterior Derivative of a Smooth Differential Form on a Smooth Manifold with Boundary

Let $n\in$ $\mathbb{N}$, let $M$ be a smooth manifold with boundary of dimension $n$ with chosen smooth atlas $((U_\alpha,\varphi_\alpha))_{\alpha\in A}$, write $\Omega_\alpha=\varphi_\alpha(U_\alpha)$, let $k\in\mathbb{N}\cup\{0\}$, and let $\omega=(\omega_\alpha)_{\alpha\in A}$ be a smooth differential $k$-form on $M$.

Fix $\alpha\in A$ and $x\in\Omega_\alpha$. Choose an open set $W$ in Euclidean space $\mathbb{R}^n$ with $x\in W$ and a differential $k$-form $\eta$ on $W$ with smooth coefficients agreeing with $\omega_\alpha$ on $W\cap\Omega_\alpha$, as in the local smoothness property of Smooth Differential k-Form on a Smooth Manifold with Boundary. Since smooth coefficient functions are in particular $C^1$ maps, $\eta$ is a $C^1$ differential $k$-form on $W$, so its exterior derivative $d\eta$ is defined. Set

$$(d\omega)_{\alpha,x}=(d\eta)_x.$$

This value does not depend on the choice of $W$ and $\eta$: the coefficient functions of any two such local forms agree on a neighborhood of $x$ that is open in the closed upper half-space $H^n$, and the first-order partial derivatives of smooth functions at a point of such a set are determined by continuity from the values of the functions on that set.

The family $((d\omega)_\alpha)_{\alpha\in A}$ so defined is a smooth differential $(k+1)$-form on $M$; its compatibility property holds because pullback commutes with the exterior derivative, by Pullback by a Smooth Map Commutes with the Exterior Derivative. This smooth $(k+1)$-form is called the exterior derivative of $\omega$ and is denoted $d\omega$.