Originally Posted by

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

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

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

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

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