Vector Calculus: Prove ∇ × (a×r/r^n) = (2-n)a/r^n + n(a.r)r/r^(n+2)

[MrWarlock616]

Homework Statement

Prove that $∇ × (\frac{\vec{a} × \vec{r}}{r^n})= \frac{2-n}{r^n}\vec{a}+\frac{n(\vec{a}.\vec{r}) \vec{r}}{r^{n+2}}$

Nothing is mentioned about $\vec{a}$ so I'm assuming it is a constant vector. Also $\vec{r} = <x, y, z>$ and $|r|=r$

Homework Equations

The Attempt at a Solution

What I don't get is why am I not allowed to do the following:

$L.H.S = \frac{1}{r^n} (∇ × (\vec{a} × \vec{r}) )$

$= \frac{1}{r^n} ((∇.\vec{r})\vec{a} - (∇.\vec{a})\vec{r} )$ ---(using vector triple product formula)

And since $∇.\vec{a} = 0$ since $\vec{a}$ is a constant and also $∇.\vec{r}=3$

$L.H.S = \frac{3\vec{a}}{r^n}$

But then a black hole gets generated somewhere in the universe because the identity cannot be proved from here.

 

[MrWarlock616]

I get it... r is not a constant. damn me

 

[Saitama]

I get it... r is not a constant. damn me

If you don't mind, can you please post the solution you have found? Thanks!

 

[MrWarlock616]

Easy way:

Using that for scalar valued point function ∅ and a vector valued point function $\vec{f}$,

$∇× (∅\vec{f}) = ∇∅ × \vec{f} + ∅ (∇ × \vec{f})$

L.H.S. $= ∇(\frac{1}{r^n}) × \vec{f} + ∅ (∇ × \vec{f})$

$= (-nr^{-n-2}\vec{r} ) × (\vec{a} × \vec{r}) + \frac{1}{r^n} (∇ × (\vec{a} × \vec{r}) )$

$= -nr^{-n-2} (\vec{r} × (\vec{a} × \vec{r})) + r^{-n} (2\vec{a})$

$= -nr^{-n-2} [(\vec{r} . \vec{r})\vec{a}-(\vec{r}.\vec{a})\vec{r}] + r^{-n} (2\vec{a})$

$= -nr^{-n-2} r^2\vec{a} + nr^{-n-2}(\vec{r} . \vec{a})\vec{r} + 2r^{-n}\vec{a}$

phew..simplifying further will yield the RHS.

Note that I've used the following:

$∇r^n=nr^{n-2} \vec{r}$

$∇ × (\vec{a} × \vec{r}) = 2\vec{a}$

Hard way:

You can do it by finding the actual cross product on the LHS like I did, keeping in mind that r is not a constant scalar xD

Also, later on you have to make an adjustment by adding and subtracting $\frac{n}{r^{n+2}} a_{1,2,3} x^2$ in each component of the cross product.

 

[MrWarlock616]

From my book

I've made many corrections to my last post

 

[Saitama]

From my book

I've made many corrections to my last post

Thanks a lot MrWarlock! :)

The alternative method is much better.