A question on notation: What does $\nabla |\vec{a} \times \vec{r}|^n$ mean? I sort of asked a version of this question before and it was unclear; try I will now to make an honest attempt to state everything clerly.
I am trying to evaluate the following, namely $\nabla w = \nabla |\vec{a} \times \vec{r}|^n$, where $\vec{a}$ is a constant vector and $\vec{r}$ is the vector $<x_1,x_2,\ldots x_n>$. Now say that I use the chain rule, first say by setting $\vec{u}$ to be equal to the cross product of $\vec{a}$ and $\vec{r}$. 
Now here's the part that I'm confused. How do we extend the chain rule over when dealing with the gradient? Do I take $\nabla |\vec{a} \times \vec{r}|^n$ to be equal to $\nabla |\vec{u}|^n$ $\times$ $\nabla (\vec{a} \times \vec{r})$, where $\times$ denotes the cross product?
The first bit $\nabla |\vec{u}|^n$ is easy, it just evaluates to $n|\vec{a} \times \vec{r}|^{n-2} (\vec{a} \times \vec {r})$, remembering that $\vec{u} = \vec{a} \times \vec{r}$.
I am guessing that $\nabla |\vec{a} \times \vec{r}|^n$ $\neq$ $\nabla |\vec{u}|^n$ $\times$ $\nabla (\vec{a} \times \vec{r})$,
as to even speak about $\nabla (\vec{a} \times \vec{r})$, i.e. the gradient of a vector we would have to talk about either the cross product or dot product of the gradient and $\nabla (\vec{a} \times \vec{r})$ 
By the way, I am told the answer given is $\nabla |\vec{a} \times \vec{r}|^n$ = $n|\vec{a} \times \vec{r}|^{n-2} \Big(\vec{a} \times (\vec{r} \times \vec{a})\Big)$.
So let's say that I try a component wise approach, i.e. we look first at $\frac{\partial w}{\partial x_1}$. Then is it true (I could be wrong) that:
$\frac{\partial w}{\partial x_1} = n|\vec{a} \times \vec{r}|^{n-2} \quad \vec{u_1} \times \frac{\partial}{\partial x_1} \Big(\vec{a} \times \vec{r}\Big) = n|\vec{a} \times \vec{r}|^{n-2} \quad \vec{u_1} \times  \Big(\vec{a} \times \frac{\partial\vec{r}}{\partial x_1}\Big)$, as $\vec{a}$ is a constant vector? Here $\vec{u_1}$ denotes the first component of the vector $\vec{a} \times \vec{r}$.
I would really aprreciate an interpretation of this, it is just that I am confused about what to take and the meanings of these operations.
 A: Let $f(r)=|a \times r|$. (I'll be lazy and omit the vector arrows.)
Then, to begin with, $\nabla(f(r)^n) = n f(r)^{n-1} \nabla f(r)$ by the chain rule. So it only remains to compute $\nabla f(r)$, and if you can't think of any clever way, then you just do it by brute force in terms of coordinates: take $a=(a_1, a_2, a_3)$ and $r=(x_1,x_2,x_3)$ (the cross products only exists in three dimensions, so your $x_n$ doesn't make sense), compute the components of $u=a \times r=(u_1,u_2,u_3)$, then compute the length $f(r)=\sqrt{u_1^2 + u_2^2 + u_3^2}$, and compute the derivatives of that with respect to $x_1$, $x_2$ and $x_3$. Done!
A: Say I differentiate $|\mathbf{a} \times \mathbf{r}|^2$. Then for the second term it agrees with $\mathbf{a} \times (\mathbf{r} \times \mathbf{a})$, but for the first term there is some confusion. Say I look at the first term of $\nabla |\mathbf{a}|^2|\mathbf{r}|^2$, namely $|\mathbf{a}|^2|\mathbf{r}|^2_{x_1} \mathbf{e_1}$ = $|\mathbf{a}|^2|\mathbf{r}|\mathbf{r}_{x_1} $ $\bullet$ $\mathbf{e_1}$? If it is indeed the dot product then I'm done! Oh by the way $\mathbf{r}_{x_1}$ means the partial derivative of $\mathbf{r}$ with respect to $x_1$.
Ben
