How to Prove the Magnetic Induction B from Vector Potential A Elegantly?

[Sagreda]

I'm having a hard time proving that if $$A=k \frac{m\times r}{(r\cdot r)^{3/2}}$$ (the vectorpotential of a magnetic dipole with moment $$m$$), then:

$$B=\nabla\times A=k\frac{3e_r(e_r\cdot m) -m}{(r\cdot r)^{3/2}}$$

without writing the whole thing in components, which becomes long, messy and ugly.
What's the elegant way of proving the magnetic induction $$B$$ in vectorial form from the vector potential $$A$$?

 

[Gerenuk]

I've attached a page from my vector notes. There you see the basic equations to treat these kind of problems. There first part I found in a book which one can check by components.

In the second part I derived some particular results from the general equations. Feel free to check them, as I haven't done that yet.

There you also find your case.