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

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?