Past exam of the mathematics course of the University of Cambridge / 2008 / ib / Paper 1 / 16B / ii / Solution

Let $s=|\mathbf{r}-{\mathbf{r}}^{\prime }|$ and introduce the magnetic vector potential $\mathbf{A}(\mathbf{r})={\mu }_0(4\pi {)}^{-1}\int \mathbf{J}({\mathbf{r}}^{\prime })/sd{V}^{\prime }$. Differentiation is with respect to $\mathbf{r}$, so $\mathbf{J}({\mathbf{r}}^{\prime })$ is constant under it. HenceTaking the curl through the integral reproduces the Biot-Savart law, proving $\boxed{\mathbf{B}=\nabla \times \mathbf{A}}$ and the requested representation.