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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?

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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.

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    $\begingroup$ But of course before "J. W. Gibbs invented vectors" they did everything in terms of coordinates. $\endgroup$
    – GEdgar
    Dec 3, 2022 at 0:45
  • $\begingroup$ According to this https://en.wikipedia.org/wiki/Divergence_theorem#History, Lagrange discovered a special case of the Divergence Theorem in 1762. Ostrogradsky proved the Divergence Theorem in 1826, and special cases were discovered and proved independently by Gauss (1813), Poisson (1824), Green (1828), and Sarrus (1828). All of these predate Gibbs, who was born in 1839. $\endgroup$ Dec 3, 2022 at 1:27
  • $\begingroup$ Forms of maximum degree are determinants and measure areas, fot that reason, the integration of forms in smooth manifold is more intuitive. So, how did they discover the "correct" definition of integrating a vector field over a surface using coordinates? $\endgroup$
    – J. López
    Dec 3, 2022 at 12:11

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