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Confusing index notation involving grad of w cross r please help

  1. Oct 14, 2012 #1
    1. The problem statement, all variables and given/known data

    consider the position vector expressed in terms of its cartesian components, r=xiei. Let w=wjej be a fixed vector whose components wj are constants that do not depend on the xi, so that δwj/δxi = 0


    2. Relevant equations

    I am trying to evaluate ∇((wXr)^2)



    3. The attempt at a solution
     
  2. jcsd
  3. Oct 14, 2012 #2

    gabbagabbahey

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    Hi troytroy, welcome to PF!:smile:

    What have you tried and where are you stuck?
     
  4. Oct 14, 2012 #3
    I am getting confused on where to begin when using index notation for these kind of problems
     
  5. Oct 14, 2012 #4

    gabbagabbahey

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    Well, here you are being asked to calulate the gradient of some scalar function, so a good place to start would be to look up the expression for the gradient of a general scalar function [itex]f[/itex] in index notation. What is that?

    Next consider that in this case, the scalar function in question is the scalar product of of a vector with itself, [itex](\mathbf{w}\times\mathbf{r})^2[/itex] (the norm-squared of a vector is usually written as [itex]||\mathbf{v}||^2[/itex], but some authors will use more clumsy notation and just call it [itex]\mathbf{v}^2[/itex]. Either way the norm-squared of a vector is given by the scalar product of a vector with itself). So, how do you express the scalar product of a vector with itself in index notation?

    Finally, consider that the vector whose norm-square you are taking the gradient of is, in this case, the cross product of a vector with another vector, [itex]\mathbf{w}\times\mathbf{r}[/itex]. How do you represent a cross product like this in index notation?
     
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