1) Show that the electromagnetic wave equation,

d^2(phi)/dx^2 + d^2(phi)/dy^2 + d^2(phi)/dz^2 –(1/c^2)( d^2(phi)/dt^2) = 0

is not invariant under Galilean transformation.

Note: here d is a partial differential operator.

I have the solution but I couldn’t understand one particular step. The solution is as follows:

The equation will be invariant if it retains the same form when expressed in terms of the new variables x’,y’,z’,t’. From Galilean transformation we have,

dx’/dx=1, dx’/dt=-v, dt’/dt=dy’/dy=dz’/dz=1, dx’/dy= dx’/dz= dy’/dx= dt’/dx=0

From chain rule and using the above results we have,

d(phi)/dx= [d(phi)/dx’][dx’/dx] + [d(phi)/dy’][dy’/dx] + [d(phi)/dz’][dz’/dx] + d(phi)/dt’][dt’/dx] = d(phi)/dx’

And,

d^2(phi)/dx^2= d^2(phi)/dx’^2

Similarly,

d^2(phi)/dy^2= d^2(phi)/dy’^2 &

d^2(phi)/dz^2= d^2(phi)/dz’^2

Moreover,

d(phi)/dt= -v[d(phi)/dx’] + d(phi)/dt’

Differentiating the above equation with respect to t ,

d^2(phi)/dt^2 = d^2(phi)/dt’^2 -2v[d^2(phi)/dx’dt’] + v^2[d^2(phi)/dx’^2]

This is where I have a doubt. I differentiated in the following way:

d^2(phi)/dt^2= -v[d^2(phi)/dx’dt] - [d(phi)/dx’][dv/dt] + [d^2(phi)/dt’^2]

= -v[(d^2(phi)/dx’^2)(dx’/dt)] + [d^2(phi)/dt’^2] ------> {dv/dt=0}

= (v^2)[ d^2(phi)/dx’^2] + [d^2(phi)/dt’^2]

I am able derive 2 of the terms but how to derive the third term -2v[d^2(phi)/dx’dt’] . Could somebody please help me with this derivation?