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Einstein summation help on a proof

  1. Mar 12, 2013 #1
    prove the identity $$\nabla\times(f\cdot\vec{v})=(\nabla f) \times \vec{v} + f \cdot \nabla \times \vec{v}$$

    I can do the proof with normal vector calculus, but I am in a tensor intensive course and would like to do this with
    einstein summation notation, but am having some trouble since I am brand new to this.

    my attempt

    left side

    [tex] \epsilon_{ijk} \partial_{j} (\nabla f \cdot \vec{v})_{k}=\epsilon_{ijk}\partial_{j} f \vec{v}_{k} [/tex]

    I didn't really know where to go from here so I moved onto the right side and expressed it in einstein notation

    [tex]\epsilon_{ijk} (\nabla f)_{j} v_{k} + f \epsilon_{ijk} \partial_{j} v_{k} [/tex]

    [tex]\epsilon_{ijk} \partial_{j} f v_{k} + f \epsilon_{ijk} \partial_{j} v_{k} [/tex]

    which I don't see how I can rearrange this to get what is on the left. I see how it could be twice what I have on the left, but that obviously is incorrect. Did I do something wrong in expressing these? Do I have to express the right side with different sets of indicees?
     
    Last edited: Mar 12, 2013
  2. jcsd
  3. Mar 12, 2013 #2

    George Jones

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    How did you go from the second last line to the last line?
     
  4. Mar 12, 2013 #3
    I just realized I forgot an f in that line
     
  5. Mar 12, 2013 #4

    George Jones

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    If you include the f, haven't you answered your own question?
     
  6. Mar 12, 2013 #5
    unless I am seeing this completely wrong, the left side (first line up above) and the right side (on the last line) is twice the left side when I add them together

    I was thinking maybe I had to express the right side like this

    [tex]\epsilon_{ijk} (\nabla f)_{j} v_{k} + f \epsilon_{klm} \partial_{l} v_{m} [/tex]

    and do the permutation identity for permutations differing by 2 indicees but I seem to be going nowhere with that
     
    Last edited: Mar 12, 2013
  7. Mar 12, 2013 #6

    George Jones

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    Apply the product rule to

    [tex]\epsilon_{ijk}\partial_{j} (f v_{k})[/tex]
     
  8. Mar 12, 2013 #7
    jeez thanks.... staring me in the face
     
  9. Mar 12, 2013 #8
    Can anyone suggest a book that has a ton of examples using einstein summation? I feel behind most of my class in regards to the notation. It just takes me too long to do problems.
     
  10. Mar 12, 2013 #9

    Fredrik

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    Just a small tip: Don't use ##\cdot## for anything other than the dot product when you're doing these things.
     
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