Curl of a vector using indicial notation

[jbrisby]

Homework Statement

A.) Show that $$\epsilon_{ijk}A_{k,j}$$ represents the curl of vector $$A_k$$

B.) Write the expression in indicial nottation:
$$\triangledown \cdot \triangledown \times A$$


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:
$$\begin{bmatrix}<br /> i &j &k \\ <br /> \frac{\partial }{\partial x_i} &\frac{\partial }{\partial x_j} &\frac{\partial }{\partial x_k} \\ <br /> A_1&A_2 &A_3 <br /> \end{bmatrix} = i(\frac{\partial A_3 }{\partial x_j}-\frac{\partial A_2 }{\partial x_k})-j(\frac{\partial A_3 }{\partial x_i}-\frac{\partial A_1 }{\partial x_k})+k(\frac{\partial A_2 }{\partial x_i}-\frac{\partial A_1 }{\partial x_j})$$

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 $$\epsilon_{ijk}A_{k,j}$$ without any mention of how that happens. If someone could break down the transformation for me it would be greatly appreciated.

 

[Quesadilla]

First of all, note that your $i,j,k$ mean different things in different equations. This could lead to some confusion.

Either way, I guess they want you to write out each component of $\nabla \times \mathbf{A}$ and $\epsilon_{ijk}A_{k,j}$ explicitly and verify that they are equal. Can you do that?

 

[jbrisby]

I edited the equation so maybe it'll make more sense. I'm not sure how to show that the cross product is transformed into the alternating tensor form.

 

[Quesadilla]

As I said, write out each component of both expressions explicitly. You have more or less done so for $\nabla \times \mathbf{A}$. It is probably more convenient to use $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$ rather than $\mathbf{i}, \mathbf{j},\mathbf{k}$.

Can you write out each component of $\epsilon_{ijk}A_{k,j}$ explicitly, i.e., do you know what the expression actually means?

 

[jbrisby]

I'm not sure that I have the grasp on the meaning of it, which is why I'm having the problem. I think that the components of the alternating tensor look like $$\begin{bmatrix}<br /> \epsilon_{111}A_{1,1} &\epsilon_{112}A_{1,2} &\epsilon_{113}A_{3,1} \\ <br /> \epsilon_{121}A_{1,2} &\epsilon_{122}A_{2,2} &\epsilon_{123}A_{3,2} \\ <br /> \epsilon_{131}A_{1,3} &\epsilon_{132}A_{3,2} &\epsilon_{133}A_{3,3}<br /> \end{bmatrix}$$

(I'm sure my notation is sloppy)

 

[Quesadilla]

Do you know what a free index and a dummy index is?

How is $\epsilon_{ijk}$ defined?

What does $A_{k,j}$ look like?

If you are having trouble answering these questions, I suggest that you read up on it in your textbook or lecture notes.