- #1

Galileo

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[tex]\vec B \times (\vec \nabla \times \vec A)[/tex]

?

I've tried it myself and found (e.g) for the x-component:

[tex]\left(B_x\frac{\partial A_x}{\partial x}+B_y\frac{\partial A_y}{\partial x}+B_z\frac{\partial A_z}{\partial x}\right)-\left(B_x\frac{\partial A_x}{\partial x}+B_y\frac{\partial A_x}{\partial y}+B_x\frac{\partial A_x}{\partial z}\right)[/tex]

I can write the last terms with the minus sign as: [itex]\vec B \cdot \nabla A_x[/itex], but I can't find a way to do something nice to the first term, except maybe:

[tex]\left(\vec B \cdot \frac{\partial}{\partial x}\vec A\right)[/tex]

I've never seen such an expression before though.

The other 2 components are similar:

[tex]\left[\vec B \times (\vec \nabla \times \vec A)\right]_y=\left(\vec B \cdot \frac{\partial}{\partial y}\vec A\right)-\left(\vec B \cdot \nabla A_y\right)[/tex]

[tex]\left[\vec B \times (\vec \nabla \times \vec A)\right]_z=\left(\vec B \cdot \frac{\partial}{\partial z}\vec A\right)-\left(\vec B \cdot \nabla A_z\right)[/tex]

I figured I may see something if I combined them all into the general expression:

[tex]\left(B_x\frac{\partial A_x}{\partial x}+B_y\frac{\partial A_y}{\partial x}+B_z\frac{\partial A_z}{\partial z}\right)\hat x +\left(B_x\frac{\partial A_x}{\partial y}+B_y\frac{\partial A_y}{\partial y}+B_z\frac{\partial A_z}{\partial y}\right)\hat y+\left(B_x\frac{\partial A_x}{\partial z}+B_y\frac{\partial A_y}{\partial z}+B_z\frac{\partial A_z}{\partial z}\right)\hat z-(\vec B \cdot \vec \nabla)\vec A[/tex]

There's definately a pattern in the first 3 terms, but the best I could come up with is writing these terms as:

[tex]B_x\nabla A_x+B_y\nabla A_y+B_z\nabla A_z[/tex]

That has condensed it a lot. Looks like a dot product with B, but....