I was studying the chapter of vector calculus in Anton´s book, more specifically the part on flux. For a parametric surface, we take the cross product of the partial derivatives of the vector equation that defines the surface with respect to each of the parameters in order to find the normal unit vector used in calculating the flux. When we have a non-parametric surface, we find the normal vector by dividing the gradient of the surface by its magnitude. I´m ok with both ideas, but my question is: why the need to separate them in two cases (parametric vs. non-parametric)? Why not take, for example, the gradient in both cases? Or the cross product?

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    $\begingroup$ To what would you apply the gradient operator in the parametric case? $\endgroup$ – amd Dec 27 '17 at 23:05
  • $\begingroup$ I don´t think I understand. $\endgroup$ – Erick3434 Dec 28 '17 at 0:18
  • $\begingroup$ In the case of a parametric surface, you’re proposing to apply the $\nabla$ operator to something, i.e., “take the gradient” of something, but what exactly is that something? Similarly, in the non-parametric case, the cross product of what with what? $\endgroup$ – amd Dec 28 '17 at 1:23

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