Which came first: Integration of vector field over the surface S or integration of forms over manifold?

Question

In integral calculus it is studied the integral of vector field over the surface S and in smooth manifold is studied the integration of forms. Let $\varphi: U\longrightarrow S$ be a parametrization of a surface $S\subset \mathbb{R}^3$ and let $F=(F_1,F_2,F_3)$ be a vector field of $\mathbb{R}^3$, so the integral of vector field over the surface is defined as $$ \int_SF=\int_U\langle F\circ \varphi,\varphi_u\wedge\varphi_v\rangle $$ where $\varphi_u=\frac{\partial{\varphi}}{\partial{u}}$ and $\varphi_v=\frac{\partial{\varphi}}{\partial v}$. On the other hand, in smooth manifold we defined the integral of forms, so if we have a vector field $X=(X_1,X_2,X_3)$ we can defined the $2$-form $$ \omega=\det(X,\cdot,\cdot)=X_1dy \wedge dz -X_2dx\wedge dz+X_3dy\wedge dz $$ and hence $$ \int_S\omega=\int_U\varphi^*\omega $$ If $X=(F_1,-F_2,F_3)$ we have the same integral. Historically, the integration the forms was first or was the vector fields?

Answer 1

Green's Theorem was developed (using differential form notation) in the early half of the nineteenth century and J. W. Gibbs invented vectors in the late nineteenth century.