# Curl of a vector using indicial notation

1. Sep 6, 2014

### jbrisby

1. The problem statement, all variables and given/known data

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

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.

Last edited: Sep 6, 2014
2. Sep 6, 2014

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?

3. Sep 6, 2014

### 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.

4. Sep 6, 2014

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?

5. Sep 6, 2014

### 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 $$\begin{bmatrix} \epsilon_{111}A_{1,1} &\epsilon_{112}A_{1,2} &\epsilon_{113}A_{3,1} \\ \epsilon_{121}A_{1,2} &\epsilon_{122}A_{2,2} &\epsilon_{123}A_{3,2} \\ \epsilon_{131}A_{1,3} &\epsilon_{132}A_{3,2} &\epsilon_{133}A_{3,3} \end{bmatrix}$$

(I'm sure my notation is sloppy)

6. Sep 6, 2014

How is $\epsilon_{ijk}$ defined?
What does $A_{k,j}$ look like?