For a given charge distribution $\rho(x, y, z)$ and divergence-free current distribution $\mathbf{J}(x, y, z)$ (i.e. $\nabla \cdot \mathbf{J}=0)$ in $\mathbb{R}^{3}$, the electric and magnetic fields $\mathbf{E}(x, y, z)$ and $\mathbf{B}(x, y, z)$ satisfy the equations

$$\nabla \times \mathbf{E}=0, \quad \nabla \cdot \mathbf{B}=0, \quad \nabla \cdot \mathbf{E}=\rho, \quad \nabla \times \mathbf{B}=\mathbf{J}$$

The radiation flux vector $\mathbf{P}$ is defined by $\mathbf{P}=\mathbf{E} \times \mathbf{B}$. For a closed surface $S$ around a region $V$, show using Gauss' theorem that the flux of the vector $\mathbf{P}$ through $S$ can be expressed as

$$\iint_{S} \mathbf{P} \cdot \mathbf{d} \mathbf{S}=-\iiint_{V} \mathbf{E} \cdot \mathbf{J} d V$$

For electric and magnetic fields given by

$$\mathbf{E}(x, y, z)=(z, 0, x), \quad \mathbf{B}(x, y, z)=(0,-x y, x z)$$

find the radiation flux through the quadrant of the unit spherical shell given by

$$x^{2}+y^{2}+z^{2}=1, \quad \text { with } \quad 0 \leqslant x \leqslant 1, \quad 0 \leqslant y \leqslant 1, \quad-1 \leqslant z \leqslant 1$$

[If you use (*), note that an open surface has been specified.]