Let $n\in$ \reftext{def:natural-numbers-2026a}{$\mathbb{N}$}, let $M$ be a \reftext{def:smooth-manifold-with-boundary-2026a}{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 \reftext{def:smooth-differential-k-form-manifold-boundary-2026a}{smooth differential $k$-form} on $M$.

Fix $\alpha\in A$ and $x\in\Omega_\alpha$. Choose an \reftext{def:open-subset-euclidean-space-2026a}{open} set $W$ in \reftext{def:euclidean-space-rn-2026a}{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 \ref{def:smooth-differential-k-form-manifold-boundary-2026a}. Since smooth coefficient functions are in particular \reftext{def:c1-map-euclidean-open-set-2026a}{$C^1$ maps}, $\eta$ is a \reftext{def:c1-differential-k-form-euclidean-open-set-2026b}{$C^1$ differential $k$-form} on $W$, so its \reftext{def:exterior-derivative-c1-differential-form-euclidean-open-set-2026b}{exterior derivative} $d\eta$ is defined. Set

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 \reftext{def:closed-upper-half-space-euclidean-2026a}{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 \reftext{def:smooth-differential-k-form-manifold-boundary-2026a}{smooth differential $(k+1)$-form} on $M$; its compatibility property holds because pullback commutes with the exterior derivative, by \ref{thm:pullback-commutes-exterior-derivative-euclidean-2026a}. This smooth $(k+1)$-form is called the \textbf{exterior derivative} of $\omega$ and is denoted $d\omega$.