1. ## vector integeral calculus

am i right in saying the following

$\phi \bigtriangledown = \bigtriangledown \phi$ where $\phi$ is some scalar field?

Im equating $\bigtriangledown\times (\phi A) = (\phi \bigtriangledown ) \times A + (\bigtriangledown \phi) \times A$

and wondering if this is the same as

$A \times (\phi A) = 2 ((\bigtriangledown \phi) \times A)$

2. Erm, isn't $A\times(\phi A) = \phi(A\times A) = 0$?

3. To be honest im really just looking for the difference between
$\phi.\bigtriangledown$ and $\bigtriangledown.\phi$

if there is any difference. I thought they would equate to the same thing?

I fixed my original post as well as i had a small but significant mistake in the formula

4. No, they are not the same thing. They are not even the same "kind" of thing.

$\nabla \phi$ is the vector function $\frac{\partial \phi}{\partial x}\vec{i}+ \frac{\partial \phi}{\partial y}\vec{j}+ \frac{\partial \phi}{\partial z}\vec{j}$.

$\phi \nabla$ is not a function at all but rather a "vector operator". If f is a real valued function then $\phi \nabla f= \phi(\frac{\partial f}{\partial x}\vec{i}+ \frac{\partial f}{\partial y}\vec{j}+ \frac{\partial f}{\partial z}\vec{k})$.

If $\vec{F}$ is a vector valued function, say $\vec{F}= f(x,y,z)\vec{i}+ g(x,y,z)\vec{j}+ h(x,y,z)\vec{k}$ then $\phi\nabla\cdot \vec{F}= \phi(\frac{\partial f}{\partial x}+ \frac{\partial g}{\partial y}+ \frac{\partial h}{\partial z})$

We could also write $\phi\nabla\times\vec{F}= \phi((\frac{\partial h}{\partial y}- \frac{\partial g}{\partial z})\vec{i}- (\frac{\partial h}{\partial x}- \frac{\partial f}{\partial z})\vec{j}+ (\frac{\partial g}{\partial x}- \frac{\partial x}{\partial y})\vec{k})$

5. Originally Posted by HallsofIvy
No, they are not the same thing. They are not even the same "kind" of thing.

$\nabla \phi$ is the vector function $\frac{\partial \phi}{\partial x}\vec{i}+ \frac{\partial \phi}{\partial y}\vec{j}+ \frac{\partial \phi}{\partial z}\vec{j}$.

$\phi \nabla$ is not a function at all but rather a "vector operator". If f is a real valued function then $\phi \nabla f= \phi(\frac{\partial f}{\partial x}\vec{i}+ \frac{\partial f}{\partial y}\vec{j}+ \frac{\partial f}{\partial z}\vec{k})$.

If $\vec{F}$ is a vector valued function, say $\vec{F}= f(x,y,z)\vec{i}+ g(x,y,z)\vec{j}+ h(x,y,z)\vec{k}$ then $\phi\nabla\cdot \vec{F}= \phi(\frac{\partial f}{\partial x}+ \frac{\partial g}{\partial y}+ \frac{\partial h}{\partial z})$

We could also write $\phi\nabla\times\vec{F}= \phi((\frac{\partial h}{\partial y}- \frac{\partial g}{\partial z})\vec{i}- (\frac{\partial h}{\partial x}- \frac{\partial f}{\partial z})\vec{j}+ (\frac{\partial g}{\partial x}- \frac{\partial x}{\partial y})\vec{k})$
Thanks a great deal for your help. I really appreciate the time you take to answer the questions so thoroughly.

Piglet