Talk:Divergence

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I think the recent changes to this article and the curl article, while correct, are not right for an audience of high-school students. With a slight change in direction, we can make the page accessible to high-school students (well, to those high-school students who can deal with partial derivatives.) The definition in terms of flux integrals divided by the volume puts it out of reach of the audience. One would have to know how surface flux integrals are defined in a coordinate-free way, and would have to be familiar with the theorems that say that this definition is independent of the surface. That requires Stokes' theorem, or the divergence theorem.

I fully agree that we want to emphasize the importance of having a "physical", "geometrical", "intrinsic", "coordinate-free" definition for the various vector operations. And, for the dot product and cross product, we can actually give that definition, and prove that it is equivalent to the algebraic definition, at the high-school level. But I don't think it is appropriate for divergence and curl.

What I'd like to do is put the rigorous geometric definition at the end of the article, in its own section. And, In the early part, say something about the importance of a coordinate-free geometric definition, but give the definition in terms of partial derivatives. Of course we have to say that this requires a Cartesian coordinate system, and say that the formulas are quite complicated in other coordinate systems. We can also suggest that, as an exercise, the reader prove that the algebraic formula is independent of Euclidean rotations. We could also say, in the later section, that a coordinate-free definition can be given in terms of tensor covariant gradients, but that that is beyond the scope of the article.

A minor nit: Saying that the "del dot V" or "del cross V" notation only applies in a Cartesian coordinate system is not true. That notation is used in all coordinates. But it is only in a Cartesian coordinate system that the algebraic formula in terms of partial derivative is correct.

The above comments also apply to the curl article. PatrickD 23:42, 22 July 2009 (EDT)

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