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$$ \overrightarrow {F}=3xz^{2}i+2xyj-x^{2}k $$ $$\phi =3x^{2}-yz $$ are given vector and scalar fields, respectively.

a) $\quad \operatorname{div}\left( \operatorname{grad}\phi \operatorname{div}\overrightarrow {F}\right) =\quad? $

b) $\quad \operatorname{curl}\left( \phi F\right) =\quad? $

c) $\quad \operatorname{div}\left( \phi F\right) =\quad? $

d) $\quad \overrightarrow {\nabla }\cdot \left( \nabla \phi \times \overrightarrow {F}\right) =\quad? $

e) $\quad \nabla \cdot \left( \overrightarrow {F}\nabla \phi \right) =\quad? $

I know the operations such as the Gradient, Divergence, Curl, and Laplacian. But I don't have an idea what can I do in this kind of problems?

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  • $\begingroup$ are all of those $x$ $y$ and $z$'s supposed to be lowercase? $\endgroup$ – orlandpm Jan 18 '13 at 0:50
  • $\begingroup$ Yes.All characters are lowercase. $\endgroup$ – Erbil Jan 18 '13 at 1:01
  • $\begingroup$ Not sure what in particular has you confused. All of these problems involve only divergences, gradients, curls, and dot or cross products, all of which you should be able to do. Can you explain in more detail what has you stuck? $\endgroup$ – Muphrid Jan 18 '13 at 17:30
  • $\begingroup$ for example : a) I found gradient of scalar field and divergence of vector field.Gradient of scalar field is a vector and divergence of vector fields is a scalar.So how can I take the divergence of this? $\endgroup$ – Erbil Jan 18 '13 at 17:39
  • $\begingroup$ Here is what I have found for a) = (18z^2-12+72x+2z-6z^2+2y-6yz) + (24-4z-3z^2-8y+2x-6xz) + (36xz+36x^2-9z^2-4y+2x-12xz-6yz-6xy) = 36x^2-6xy-12yz+6y+18xz $\endgroup$ – Erbil Jan 18 '13 at 20:04

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