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Gradient of dot product using suffix notation

  1. Feb 12, 2016 #1
    1. The problem statement, all variables and given/known data
    Find the gradient of [tex]\underline{\nabla}(\underline{a}\cdot\underline{r})^n[/tex] where a is a constant vector, using suffix notation and chain rule.
    2. Relevant equations
    On the previous problem,s I found that grad(a.r)=a and grad(r)=[itex]\underline{\hat{r}}[/itex]

    3. The attempt at a solution
    [tex]
    \underline{e_i}(\frac{\partial }{\partial x_i})(\underline{a}\cdot\underline{r})^n=\underline{e_i}(\frac{\partial }{\partial x_i}(a_jx_j)^n)=\underline{e_i}(n(a_jx_j)^{n-1}(a_j\delta_{ij}))
    [/tex]
    I'm sure that the last step is wrong so could someone lead me to the right direction?
    Thank you!
     
    Last edited: Feb 12, 2016
  2. jcsd
  3. Feb 12, 2016 #2

    HallsofIvy

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    The derivative of [itex](ax)^n[/itex] is NOT [itex]n(ax)^{n-1}[/itex].
     
  4. Feb 12, 2016 #3
    That is what I thought. Could you explain that step?
     
  5. Feb 12, 2016 #4

    HallsofIvy

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    Just after I wrote that I noticed that you also had "[itex]a_j\delta_{ij}[/itex]". The derivative of [itex](ax)^n= a^nx^n[/itex] is [itex]na^nx^{n-1}[/itex] but that can also be written as [itex](ax)^{n-2}(a)[/itex] where the last "a" is due to the chain rule. Was that what you meant?
     
  6. Feb 12, 2016 #5

    BvU

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    ##na^nx^{n-1}## is confusing. The inner product ##(a_j x_j)^{n-1}## is lost.

    When I write it out and then put it back together again ##
    \underline{e_i}(n(a_jx_j)^{n-1}(a_j\delta_{ij}))## seems OK.
    ##
    na_i \underline{e_i}\;(a_jx_j)^{n-1}## might be somewhat more elegant,
    and even that can be simplified further !
     
    Last edited: Feb 12, 2016
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