Source-free electromagnetic fields exist in a localized region of space. Consider the various conservation laws that are contained in the integral of ∂_αM^αβγ = 0 over all space, where M^αβγ is defined by (12.117).  

(a) Show that when β and γ are both space indices conservation of the total field angular momentum follows.

(b) Show that when β = 0 the conservation law is

dX/dt = c²P_em/E_em

where X is the coordinate of the center of mass of the electromagnetic fields, defined by

X ∫ ud³x = ∫ xu d³x

where μ is the electromagnetic energy density and E_em and P_em are the total energy and momentum of the fields.