Jackson: eq 5.21

  • Thread starter shehry1
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Can anyone explain to me how Jackson goes from equation 5.20 to 5.21 (Magnetostatics - Derivation of the curl of B in terms of the current density).

He says that he's used integration by parts but I cant see how he got rid of the first term (the one that involves integrals only) when integrating by parts.

Regards
 

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  • #2
gabbagabbahey
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It looks like he uses the vector identity [tex]\mathbf{A} \cdot (\mathbf{\nabla}f)=\mathbf{\nabla} \cdot (f \mathbf{A})-f(\mathbf{\nabla}\cdot \mathbf{A})[/tex] with [tex]f=\frac{1}{|\mathbf{x}-\mathbf{x}'|}[/tex] and [tex]\mathbf{A}=\mathbf{J}(\mathbf{x}')[/tex]

The [tex]\int \mathbf{\nabla}' \cdot \frac{\mathbf{J}(\mathbf{x}')}{|\mathbf{x}-\mathbf{x}'|} d^3x'[/tex] term can be transformed to a surface integral using the divergence theorem, and since the boundary of all space is at [tex]x' \to \infty[/tex] the integrand of that term is zero leaving you with equation 5.21
 

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