Wikipedia:Reference desk/Archives/Mathematics/2011 January 21

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January 21

Vector calculus question

Why is (A*del)B equal to A(del*B)? Why is A*del not equal to del*A? --140.180.26.37 (talk) 06:14, 21 January 2011 (UTC)

Do you mean $(A\times \nabla )\cdot B=A\cdot (\nabla \times B)$? This can be remembered because it follows the general pattern of the scalar triple product, but to prove it properly, you'd have to write them out in coordinates and observe that they turn out to be the same. This is slightly tedious but not hard.

$A\times \nabla$ is different from $\nabla \times A$ (above and beyond merely being minus each other, as usual for cross products) because that's the way the notation work. More precisely, the convention that differential operators always operate on the thing to the right of them and do not care about things to their left. So $\nabla \times A$ is just a vector field, but $A\times \nabla$ is a differential operator still "hungry for something to differentiate" (that is, same kind of thing as $\nabla$ itself) because there is nothing to the right of the $\nabla$ for it to operate on.

There's no deep theoretical reason it has to be that way; the convention just makes it easier to remember how the notation works. –Henning Makholm (talk) 08:12, 21 January 2011 (UTC)

No, by *, I meant the dot product and not the cross product. --140.180.26.37 (talk) 20:31, 21 January 2011 (UTC)

But it is not true that $(A\cdot \nabla )B=A(\nabla \cdot B)$. As a counterexample, set A=(1,0,0) and B(x,y,z)=(0,x,z). Then,

$(A\cdot \nabla )B=(1{\frac {\partial }{\partial x}}+0{\frac {\partial }{\partial y}}+0{\frac {\partial }{\partial z}})B={\frac {\partial }{\partial x}}B=(0,1,0)$ but $A(\nabla \cdot B)=A({\frac {\partial }{\partial x}}0+{\frac {\partial }{\partial y}}x+{\frac {\partial }{\partial z}}z)=A(0+0+1)=(1,0,0)$

assuming that $A(\nabla \cdot B)$ means the scalar multiple of A by the divergence of B. –Henning Makholm (talk) 21:43, 21 January 2011 (UTC)

Using coordinate calculations doesn't always work. If the tangent bundle is a trivial bundle, i.e. a product, then you're okay. But if not then coordinate calculations don't work. For example, locally, every closed differential form is an exact differential form (by the Poincaré lemma). But this isn't always true globally; it depends on the topology of the manifold. — Fly by Night (talk) 04:27, 22 January 2011 (UTC)

If the identity isn't even true in Euclidean space, there is no chance it will work in arbitrary manifolds. (And I don't really see any indication that the OP is thinking about manifolds at all). –Henning Makholm (talk) 22:35, 22 January 2011 (UTC)