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jbrisby
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Homework Statement
A.) Show that [tex]\epsilon_{ijk}A_{k,j} [/tex] represents the curl of vector [tex] A_k [/tex]
B.) Write the expression in indicial nottation:
[tex]\triangledown \cdot \triangledown \times A[/tex]
2. The attempt at a solution
I'm hoping that if I can get help on part A.) it will shed light on part B.) I have several more of these to do but not going to ask all of them here. For A.) I have done the cross product easily enough:
[tex]\begin{bmatrix}
i &j &k \\
\frac{\partial }{\partial x_i} &\frac{\partial }{\partial x_j} &\frac{\partial }{\partial x_k} \\
A_1&A_2 &A_3
\end{bmatrix} = i(\frac{\partial A_3 }{\partial x_j}-\frac{\partial A_2 }{\partial x_k})-j(\frac{\partial A_3 }{\partial x_i}-\frac{\partial A_1 }{\partial x_k})+k(\frac{\partial A_2 }{\partial x_i}-\frac{\partial A_1 }{\partial x_j})[/tex]
I'm having problems transforming this into the alternating tensor form. Everything I've found for the problem just states that the product can be expressed as [tex]\epsilon_{ijk}A_{k,j}[/tex] without any mention of how that happens. If someone could break down the transformation for me it would be greatly appreciated.
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