Vector potential and magnetic induction

Question

Given is a vector potential

$$A(\vec{x},t)= \frac{\mu_0}{4\pi}\frac{\vec{m}\times\vec{x}}{|\vec{x}|^3}$$

Now I want to calculate the magnetic induction $\vec B$: $$\vec{B} = \nabla\times{\vec{A}} = \frac{\mu_0}{4\pi}\left(\nabla\times (\vec{m}\times\vec{x})\frac{1}{|\vec{x}|^3}+\nabla\left(\frac{1}{|\vec{x}|^3}\right)\times(\vec{m}\times\vec{x})\right)$$

Can someone please explain me how to get the grad term in this equation?

Answer 1

You have a scalar $\frac{\mu_0}{4\pi|\vec{x}|^3}\equiv k$ multiplied by the cross product of two vectors $\vec{m}\times\vec{x}\equiv \vec{z}$. There is a vector identity:

$$\nabla\times(k\vec{z})=k(\nabla\times\vec{z})+(\nabla k)\times\vec{z}$$