I am currently discovering vector analysis and was asked to prove that $(\vec{u} . \nabla) f \neq ( \nabla . \vec{u})f$ with $\vec{u}$ a vector field and $f$ a scalar field. I really struggle to understand how this nabla thing works. Can someone give me examples/rules ?
I found the correct proof but I don't understand why it works :
$(\vec{u} . \nabla) f = (u_x \partial_x + u_y \partial_y + u_z \partial_z)f $ but why is $u_x \partial_x \neq \partial_x u_x$, and what is $u_x \partial_x$, why is this "$\partial_x$" not containing anything inside ? Shouldn't it be differentiating a function ? Sorry for all these questions, this is very new to me.
Best Answer
"Nabla" is a symbolic "vector differential operator". It can be written, symbolically, $\nabla= \frac{\partial}{\partial x}\vec{i}+ \frac{\partial}{\partial Y}\vec{j}+ \frac{\partial}{\partial z}\vec{k}$.
Just as there are three kinds of vector products, "scalar product", which multiplies a vector and a scalar to get a vector, "dot product", which multiplies two vectors to get a scalar, and "cross product", which multiplies two vectors to get a vector, so there are three kinds of "products" with nabla.
"grad" (gradient) which applies to a scalar valued function, f(x, y, z), to get a vector function: $\nabla f= (\frac{\partial}{\partial x}\vec{i}+ \frac{\partial}{\partial y}\vec{j}+ \frac{\partial}{\partial z}\vec{k})f(x,y,z)= \frac{\partial f}{\partial x}\vec{i}+ \frac{\partial f}{\partial y}\vec{j}+ \frac{\partial f}{\partial z}\vec{k}$"
"div" (divergence) which applies to a vector valued function, $\vec{F(x,y,z)}= f(x,y,z)\vec{i}+ g(x,y,z)\vec{j}+ h(x,y,z)\vec{k}$ to get a scalar valued function: $\nabla\cdot \vec{F}= \frac{\partial f}{\partial x}+ \frac{\partial g}{\partial y}+ \frac{\partial h}{\partial z}$.
"curl" which applies to a vector valued function, $\vec{F(x,y,z)}= f(x,y,z)\vec{i}+ g(x,y,z)\vec{j}+ h(x,y,z)\vec{k}$, to get a vector valued function: $\nabla\times F= $$\left|\begin{array}{ccc} \vec{i} & \vec{j} & \vec{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ f(x,y,z) & g(x,y,z) & h(x,y,z)\end{array}\right|$
$= \left(\frac{\partial h}{\partial y}- \frac{\partial g}{\partial z}\right)\vec{i}+ \left(\frac{\partial f}{\partial z}- \frac{\partial h}{\partial x}\right)\vec{j}+ \left(\frac{\partial g}{\partial x}- \frac{\partial f}{\partial y}\right)\vec{k}$.