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JHansen 2007-11-19 11:04

Formula split up
 
I'm doing a paper on the Hodge theorem and as part of this, I'm reading an article that presents the ideas of Hodge decomposition in the special case of domains in [tex]R^3[/tex] with a smooth boundary.

In order to prove certain propositions, the author uses the results from electrodynamics about how the magnetic field from a current distribution looks like. The way he presents the Maxwell law is a bit puzzling to me. The way I'm used to see Maxwell's law is
[CENTER][tex] \nabla\times B = J+\frac{\partial E}{\partial t},[/tex][/CENTER]
but the author uses a slightly different version. He writes:
[CENTER][tex] \nabla_y\times BS(V)(y) = \left\{ \begin{tabular}{ll}V(y) & \text{for }y\in\Omega \\ 0 & \text{for }y\in\Omega' \end{tabular}\right\} + \frac{1}{4\pi}\nabla_y\int_\Omega\frac{\nabla_x\cdot V(x)}{|y-x|}d(\text{vol}_x) - \frac{1}{4\pi}\nabla_y\int_{\partial\Omega}\frac{V(x)\cdot n}{|y-x|}d(\text{area}_x),[/tex][/CENTER]
where BS(V)(y) is the magnetic field from a current flow V, at the position y, [tex]\Omega[/tex] is our domain with smooth boundary, [tex]\Omega'[/tex] is the closure of [tex]R^3-\Omega[/tex], [tex]\nabla_x[/tex] means differentiation w.r.t. x (and likewise for y), and [I]n[/I] is a boundary unit normal vector.

Now to my question: It is clear that the term coming from a changing electric field has been broken into two parts: a part that accounts for that which happens 'inside' the domain and a part that accounts for that which happens at the boundary of the domain. Is this a purely mathematical trick that you could do with any vector field, or is there some physics at heart of this that allows the author to do this?

Any hints as to how this split up is done would be greatly appreciated, as is a literature reference.

--
Best regards,
Jes

davieddy 2007-11-20 11:42

The printing is clear for me too. But who in their
right mind would use "V" to denote current???

davieddy 2007-11-20 13:30

[quote=davieddy;118861]The printing is clear for me too. But who in their
right mind would use "V" to denote current???[/quote]
Unless he has "An exceptionally simple" explanation of how
the two may be related.

JHansen 2007-11-20 14:55

[QUOTE=davieddy;118863]Unless he has "An exceptionally simple" explanation of how
the two may be related.[/QUOTE]It comes from the way the author uses the Biot-Savart law. Given a vector field V, he looks at the magnetic field BS(V) that would be generated if V was a currentflow.

davieddy 2007-11-20 15:02

[quote=JHansen;118866]It comes from the way the author uses the Biot-Savart law. Given a vector field V, he looks at the magnetic field BS(V) that would be generated if V was a currentflow.[/quote]
THX for the explanation. But in my book V = "potential" (voltage)


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