- #1

- 305

- 2

## Main Question or Discussion Point

In Jackson's 'classical electrodynamics' he re-expresses a volume integral of a vector in terms of a moment like divergence:

[tex]\begin{align}\int \mathbf{J} d^3 x = - \int \mathbf{x} ( \boldsymbol{\nabla} \cdot \mathbf{J} ) d^3 x\end{align} [/tex]

He calls this change "integration by parts". If this is integration by parts, there must be some form of chain rule (where one of the terms is zero on the boundry), but I can't figure out what that chain rule would be. I initially thought that the expansion of

[tex]\begin{align}\boldsymbol{\nabla} (\mathbf{x} \cdot \mathbf{J})\end{align} [/tex]

might have the structure I was looking for (i.e. something like [itex]\mathbf{x} \boldsymbol{\nabla} \cdot \mathbf{J}+\mathbf{J} \boldsymbol{\nabla} \cdot \mathbf{x}[/itex]), however

[tex]\begin{align}\boldsymbol{\nabla} (\mathbf{x} \cdot \mathbf{J}) =\mathbf{x} \cdot \boldsymbol{\nabla} \mathbf{J}+\mathbf{J} \cdot \boldsymbol{\nabla} \mathbf{x}+ \mathbf{x} \times ( \boldsymbol{\nabla} \times \mathbf{J} )= \mathbf{J} + \sum_a x_a \boldsymbol{\nabla} J_a.\end{align} [/tex]

I tried a few other gradients of various vector products (including [itex]\boldsymbol{\nabla} \times ( \mathbf{x} \times \mathbf{J} )[/itex]), but wasn't able to figure out one that justifies what the author did with this integral.

I am probably missing something obvious (or at least something that is obvious to Jackson) ?

[tex]\begin{align}\int \mathbf{J} d^3 x = - \int \mathbf{x} ( \boldsymbol{\nabla} \cdot \mathbf{J} ) d^3 x\end{align} [/tex]

He calls this change "integration by parts". If this is integration by parts, there must be some form of chain rule (where one of the terms is zero on the boundry), but I can't figure out what that chain rule would be. I initially thought that the expansion of

[tex]\begin{align}\boldsymbol{\nabla} (\mathbf{x} \cdot \mathbf{J})\end{align} [/tex]

might have the structure I was looking for (i.e. something like [itex]\mathbf{x} \boldsymbol{\nabla} \cdot \mathbf{J}+\mathbf{J} \boldsymbol{\nabla} \cdot \mathbf{x}[/itex]), however

[tex]\begin{align}\boldsymbol{\nabla} (\mathbf{x} \cdot \mathbf{J}) =\mathbf{x} \cdot \boldsymbol{\nabla} \mathbf{J}+\mathbf{J} \cdot \boldsymbol{\nabla} \mathbf{x}+ \mathbf{x} \times ( \boldsymbol{\nabla} \times \mathbf{J} )= \mathbf{J} + \sum_a x_a \boldsymbol{\nabla} J_a.\end{align} [/tex]

I tried a few other gradients of various vector products (including [itex]\boldsymbol{\nabla} \times ( \mathbf{x} \times \mathbf{J} )[/itex]), but wasn't able to figure out one that justifies what the author did with this integral.

I am probably missing something obvious (or at least something that is obvious to Jackson) ?