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Homework Help: Curl of a vector using indicial notation

  1. Sep 6, 2014 #1
    1. The problem statement, all variables and given/known data

    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:
    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.
    Last edited: Sep 6, 2014
  2. jcsd
  3. Sep 6, 2014 #2
    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?
  4. Sep 6, 2014 #3
    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.
  5. Sep 6, 2014 #4
    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?
  6. Sep 6, 2014 #5
    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 [tex] \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} [/tex]

    (I'm sure my notation is sloppy)
  7. Sep 6, 2014 #6
    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.
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