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Why must this expression for the curl be wrong?

  1. May 4, 2017 #1
    1. The problem statement, all variables and given/known data

    Without explicit calculation, argue why the following expression cannot be correct: $$\nabla \times (\mathbf{c} \times \mathbf{r}) = c_{2}\mathbf{e_{1}}+c_{1}\mathbf{e_{2}}+c_{3}\mathbf{e_{3}}$$ where ##\mathbf{c}## is a constant vector and ##\mathbf{r}## is the position vector.

    2. Relevant equations


    3. The attempt at a solution

    So I can do the explicit calculation to see that in fact the curl should be parallel to the vector ##\mathbf{c}## but then I struggle to provide an argument for why this should be so without the calculation.

    I think that the incorrect solution has flipped the vector ##\mathbf{c}## in the x-y plane but left the z component unchanged. The position vector treats all directions equally so it seems strange that the z-component of ##\mathbf{c}## should be unchanged by this operation. However, I am unable to explain why this solution can't be true.
     
  2. jcsd
  3. May 4, 2017 #2

    BvU

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    Hi,
    Can you show us in detail ? What does 'parallel' mean to you ?

    Are you allowed to use / familiar with the triple product expansion ?
     
  4. May 4, 2017 #3
    Hi,

    So I used the formula ##\nabla \times(\mathbf{c}\times\mathbf{r}) = (\nabla \cdot \mathbf{r})\mathbf{c}+(\mathbf{r}\cdot\nabla)\mathbf{c}-(\nabla\cdot\mathbf{c})\mathbf{r}-(\mathbf{c}\cdot\nabla)\mathbf{r}##. Then the terms where ##\nabla## acts on ##\mathbf{c}## will be zero since ##\mathbf{c}## is constant. Also ##\nabla \cdot \mathbf{r}=3## and ##(\mathbf{c}\cdot\nabla)\mathbf{r}=\mathbf{c}## so the whole expression reduces to ##3\mathbf{c}-\mathbf{c}=2\mathbf{c}## which is why I though that the answer should then be parallel to ##\mathbf{c}##.

    However, I think the point of the question was to justify this intuitively without explicitly doing the calculation above. And that is where I'm unsure.

    Thanks
     
  5. May 4, 2017 #4

    BvU

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    I see. 'Parallel' in the sense of 'linearly dependent'.
    I was under the impresssion you worked out the components of ##\vec c \times\vec r## and then applied the ##\vec \nabla \times ## to the result. That, to me, is an explicit calculation. I tried it and I think it yields ##2\,\vec c## as you found.

    So you are fine.

    However, with the triple product expansion expression in the link I gave, I managed to confuse myself: the Lagrange formula reads $${\bf a}\times\left ( {\bf b} \times {\bf c} \right ) = {\bf b} \left ( {\bf a} \cdot {\bf c} \right ) - {\bf c} \left ( {\bf a} \cdot {\bf b} \right ) $$so that $$
    \nabla \times(\mathbf{c}\times\mathbf{r}) = \mathbf{c} (\nabla \cdot \mathbf{r}) - \mathbf{r} (\nabla\cdot \mathbf{c}\ ) \ ,$$ only two terms, and yielding ##3\bf c##....:woot:

    Perhaps some math expert can put me right ?
     
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