Hi, I have a book that makes the equality.(adsbygoogle = window.adsbygoogle || []).push({});

[itex] \vec{B}dV = (\vec{e_1}B_1 + \vec{e_2}B_2 + \vec{e_1}B_2)dx_1 dx_2 dx_3 \\[1ex]

= dx_1 \vec{e}_1(B_1 dx_2 dx_3 ) + dx_2 \vec{e}_2(B_2 dx_1 dx_3 ) + dx_3 \vec{e}_3 (B_3 dx_1 dx_2) = (\vec{B}\cdot d\vec{S}) d\vec{l}. [/itex]

I'm a bit confused as to how it makes that last equality. In a very general sense, the surface element is given by;

[itex] d\vec{S} = (dx_2dx_3,dx_1dx_3,dx_1dx_2) [/itex]

right? What I need is a way of represententing [itex] d\vec{l} = (dx_1,dx_2,dx_3)[/itex] as being multiplied component-wise by the 3 summation terms of [itex] \vec{B} \cdot d\vec{S} [/itex], but as far as I can tell the notation [itex] (\vec{B}\cdot d\vec{S})d\vec{l} [/itex] doesn't seem to do that?

If this is not possible, it might be cause it's specific to my situation. I'm looking at the integral of [itex] B [/itex] over the volume a plasma flux rope - which is defined as the volume encompassed by a fixed selection of magnetic field lines.

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# Volume integral turned in to surface + line integral?

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