Coulomb gauge derivation (static field)

[kidsasd987]

Hi, I am a little confused of derivation of Coulomb Gauage.

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First, prime notation is adopted to describe the magnetic field density source current.
Non-prime notation is for position that we are specifically interested in (ex. the position magnetic force acts on).

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if equation (13) is true, then divergence of vector potential is 0.Hence,

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(14)we can use vector indentity (14) to express (13) in another form.

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at this step, I am lost. We can break down divergence of vector potential into
two parts. I understand the first part is 0 because we assume that the field is static with respect to time,
but I wonder why the second flux integral part is 0.
(reference: http://ghebook.blogspot.ca/2011/06/energy-of-magnetic-field.html)

 

[vanhees71]

I can't read the website, but the 2nd integral should be 0 since the volume must enclose the full region, where the current density is non-zero, i.e., at the boundary of the volume $\vec{J}=0$, and thus the 2nd integral vanishes. The first integral vanishes, because the integrability condition for the magnetostatic field is $\vec{\nabla} \cdot \vec{J}=0$.

The integrability condition for the full Maxwell equations is the continuity equation for the current density, i.e.,
$$\partial_t \rho + \vec{\nabla} \cdot \vec{J}=0,$$
which is the local form of electric-charge conservation and a consequence of the gauge invariance of electromagnetics.