>>10977297The integral on the right is adding up volume elements. However take a point charge, the divergence of it's electric field is zero everywhere except the location of the charge itself. Therefore, since we are modelling our extended charge as a sum of differentially small point charges, we can say that any volume element without a charge will contribute zero to the final integral. However, wherever there exists a charge, the volume element will contribute , where is the density of charge at that location.
We do something quite similar for magnetism. Since the magnetic field around an infinite wire is proportional to 1/r, and the circumference of a circle around the wire is proportional to r, we get something called Ampere's law:
where "I" is the current enclosed in the amperian loop. This is not a surface integral such as we had before, but rather a line integral. This time we are doing a dot-product with a line differential line element running around the loop.
Once again, we can use vector calculus to change the form of this integral, turning it into a surface integral.
And once again, we can use the same reasoning, breaking this thing up into little differentially small current elements which run through our surface. The curl will be "0" everywhere there is no current, but everywhere there is current the curl will be proportional to the current density .
Since the divergence of a curl is always zero, we can also say that everywhere.
