Because is impossible to calculate the antiderivative of a function of
more than one variable, indefinite multiple integrals do not exist so
they are all definite integrals.

http://en.wikipedia.org/wiki/Multiple_integral


Currents, i.e. moving electric charges, produce magnetic fields. There
are no magnetic charges. Maxwell's equations tell us how to compute
the electric fields and magnetic field produced by charged particles.
The terms electrostatics and magnetostatics refer to steady state
conditions, when all charges are at rest or only steady currents are
flowing. Then the charge densities do not change anywhere. Under
those conditions Maxwell's equations are given below...

http://electron9.phys.utk.edu/phys13.../m7/Ampere.htm


In Section V we derive the electric and magnetic fields
from the Coulomb-gauge potentials
and show that they are the well known expressions,
causal and propagating with speed c, despite
the instantaneous nature of the scalar potential.
This ground has been traveled before in this
journal by Brill and Goodman6 and recently by Rohrlich.7
There is also Problem 6.20 in my
book.2 Our discussion here is different and I think
more transparent because of the form of our
solution for AC. Some aspects of Brill and Goodman come
close. In Section VI we discuss
briefly the quasi-static limit of the vector potential in
the Coulomb gauge and its use to obtain a
Lagrangian for the interaction of charged particles that
is correct to order 1/c2 in the velocities.
Section VII is devoted to a class of gauges we call the
velocity gauge (v-gauge) in which the scalar
potential propagates with an arbitrary speed v.
The Lorenz and Coulomb gauges are limiting
cases, v = c and v = , respectively. The gauge function
and the potentials are determined, as are
the electromagnetic fields (the same as always).

http://arxiv.org/abs/physics/0204034



Sue...