Formula split up

JHansen · 2007-11-19, 11:04

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 $R^3$ 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

$\nabla\times B = J+\frac{\partial E}{\partial t},$

but the author uses a slightly different version. He writes:

$\nabla_y\times BS(V)(y) = \left\{ \begin{tabular}{ll}V(y) & \text{for }y\in\Omega \\ 0 & \text{for }y\in\Omega' \end{tabular}\right\} + \frac{1}{4\pi}\nabla_y\int_\Omega\frac{\nabla_x\cdot V(x)}{|y-x|}d(\text{vol}_x) - \frac{1}{4\pi}\nabla_y\int_{\partial\Omega}\frac{V(x)\cdot n}{|y-x|}d(\text{area}_x),$

where BS(V)(y) is the magnetic field from a current flow V, at the position y, $\Omega$ is our domain with smooth boundary, $\Omega'$ is the closure of $R^3-\Omega$ , $\nabla_x$ means differentiation w.r.t. x (and likewise for y), and n 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:

Originally Posted by davieddy

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

Unless he has "An exceptionally simple" explanation of how
the two may be related.

JHansen · 2007-11-20, 14:55

Quote:

Originally Posted by davieddy

Unless he has "An exceptionally simple" explanation of how
the two may be related.

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:

Originally Posted by JHansen

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.

THX for the explanation. But in my book V = "potential" (voltage)

2007-11-19, 11:04	#1
JHansen Apr 2004 Copenhagen, Denmark 2²×29 Posts	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 $R^3$ 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 $\nabla\times B = J+\frac{\partial E}{\partial t},$ but the author uses a slightly different version. He writes: $\nabla_y\times BS(V)(y) = \left\{ \begin{tabular}{ll}V(y) & \text{for }y\in\Omega \\ 0 & \text{for }y\in\Omega' \end{tabular}\right\} + \frac{1}{4\pi}\nabla_y\int_\Omega\frac{\nabla_x\cdot V(x)}{\|y-x\|}d(\text{vol}_x) - \frac{1}{4\pi}\nabla_y\int_{\partial\Omega}\frac{V(x)\cdot n}{\|y-x\|}d(\text{area}_x),$ where BS(V)(y) is the magnetic field from a current flow V, at the position y, $\Omega$ is our domain with smooth boundary, $\Omega'$ is the closure of $R^3-\Omega$ , $\nabla_x$ means differentiation w.r.t. x (and likewise for y), and n 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

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2007-11-20, 11:42	#2
davieddy "Lucan" Dec 2006 England 2×3×13×83 Posts	The printing is clear for me too. But who in their right mind would use "V" to denote current???