JE Brock found the centroid of a spherical triangle $T=\triangle ABC$ with area $[T]$ to be
$$g=\frac{1}{2[T]}\left(\frac{A\times B}{|A\times B|}c+\frac{B\times C}{|B\times C|}a+\frac{C\times A}{|C\times A|}b\right)$$
Thinking of $A\times B/|A\times B|$ as the unit vector perpendicular to side $c$, we can write this as
$$\int_T \vec{r}dS=g*[T]=\frac{1}{2}\int_{\partial T} \vec{u} ds$$
where $\vec{u}$ is the inward pointing unit vector and $S$ is uniform measure on the sphere.
Question: This formulation looks like the Stokes or divergence theorem. Is there a way to quickly get Brock's result with a fundamental theorem of calculus? I'm hoping to analyze centroids in higher dimensions.