Derivation of Maxwells equations from Coulombs and Biot-Savart laws, Lorentz force and continuity equation



div(E)=(4pi*e)-1(rho)(div)s^/s2dV =(4pi*e)-1(rho)(div)s^/s24pi(delta(s))dV

div(E)=(rho)/e

Where e is epsilon-naught (the permitivity of free space)
the fact that rho is a function of s' and the the differentials are of s has been used.
convince yourself that div(s^/s2)=4pi(delta(s))

in media (div)E=(rho)/e=e-1((rhofree)+(rhobound)) =(rho)/e=e-1((rhofree)-div(P))
rearanging gives
div(eE+P)=rhofree
that is

div(D)=rhofree



(div)B=(u/4pi)(div)(JXs^/s2)dV =(u/4pi)s^/s2(curl)(J)-J(curl)(s^/s2)dV

s^/s2 is not the least bit curly!
So, both the therms of the integrand are zero, always.
(div)B=0



(curl)E=(4pi*e)-1(rho)(curl)s^/s2dV =0

But
(curl)E*dA=loopE*dl
is not in general zero. If it where, there could never be electrically driven currents!
Consider the magnetic flux through a loop
d(phi)=B*dA=loopB*vdtXdl
thus
d(phi)/dt=loop-vXB*dl
We know vXB to be a force per unit charge. We can no longer think of E as it was defined in coulombs law, it is a more general force per charge. Bu thow is this force at a distance being exerted? Let's just blame it on the electric field (better explanation needed)
d(phi)/dt=-loopE*dl
which can be written
((d/dt)B*dA=-(curl)E*dA
It can be seen now
(curl)E=-dB/dt


(curl)B=(u/4pi)(curl)(JXs^/s2)dV

curl(AXB)=B(div)C-B(div)C-(C(div)B-C(div)B)
so

(curl)B=(u/4pi)J(div)s^/s2-J(div)s^/s2-(s^/s2(div)J-s^/s2(div)J)dV

Again, the derivatives of J are zero because J is not a function of s. So...

(curl)B=(u/4pi)J(div)s^/s2-J(div)s^/s2dV=uJ-(u/4pi)J*(div)s^/s2dV
Consider each component of s^/s2 in (J*(div))s^/s2 to be fi
div(Jfi)=fidiv(J)+J*grad(fi)
so
(curl)B=uJ-(u/4pi)div(Jfi)- fidiv(J)dV
=uJ+(u/4pi)div(Jfi)dV
=uJ+(u/4pi)loopJfi*dA
The surface can be any surface which encloses the volume. At infinity the value of the integrand is zero, so the integral must be zero everywhere.
(curl)B=uJ

This cant be the whole story because the divergence of the curl of any function must be zero
div(curl(B))=u*div(J)
by the continuity equation
div(curl(B)=-u*d(rho)/dt
and by the first of Maxwell's equations above
div(curl(B)=-u*d(e*div(E))/dt=-u*e*div(dE/dt)
To force the divergence of the curl of the E field to be zero, we add the term u*e*dE/dt

curl(B)=uJ+u*e*dE/dt

in matter
curl(B)=u(Jf+Jbound+dP/dt)+u*e*dE/dt =u(Jf+curl(M)+dP/dt)+u*e*dE/dt

where dP/dt is the current generated by the bound charge changing
{P/dt=dI/daperp implies dI=P*daperp/dt=d(sigmabound)da/dt}

rearanging gives
curl(B/u-M)=Jf+d(eE+P)/dt
That is

curl(H)=Jf+dD/dt


knowing is half the battle