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Vector calculus -.-

  1. Nov 19, 2013 #1
    1. The problem statement, all variables and given/known data

    Prove that ##∇ × (\frac{\vec{a} × \vec{r}}{r^n})= \frac{2-n}{r^n}\vec{a}+\frac{n(\vec{a}.\vec{r}) \vec{r}}{r^{n+2}}##

    Nothing is mentioned about ##\vec{a}## so I'm assuming it is a constant vector. Also ##\vec{r} = <x, y, z> ## and ##|r|=r##

    2. Relevant equations



    3. The attempt at a solution

    What I don't get is why am I not allowed to do the following:

    ##L.H.S = \frac{1}{r^n} (∇ × (\vec{a} × \vec{r}) ) ##

    ## = \frac{1}{r^n} ((∇.\vec{r})\vec{a} - (∇.\vec{a})\vec{r} )## ---(using vector triple product formula)

    And since ##∇.\vec{a} = 0## since ##\vec{a}## is a constant and also ##∇.\vec{r}=3##

    ##L.H.S = \frac{3\vec{a}}{r^n} ##

    But then a black hole gets generated somewhere in the universe because the identity cannot be proved from here.
     
  2. jcsd
  3. Nov 19, 2013 #2
    I get it... r is not a constant. damn me
     
  4. Nov 19, 2013 #3
    If you don't mind, can you please post the solution you have found? Thanks!
     
  5. Nov 19, 2013 #4
    Easy way:

    Using that for scalar valued point function ∅ and a vector valued point function ##\vec{f}##,

    ##∇× (∅\vec{f}) = ∇∅ × \vec{f} + ∅ (∇ × \vec{f})##

    L.H.S. ## = ∇(\frac{1}{r^n}) × \vec{f} + ∅ (∇ × \vec{f}) ##

    ## = (-nr^{-n-2}\vec{r} ) × (\vec{a} × \vec{r}) + \frac{1}{r^n} (∇ × (\vec{a} × \vec{r}) )##

    ## = -nr^{-n-2} (\vec{r} × (\vec{a} × \vec{r})) + r^{-n} (2\vec{a}) ##

    ## = -nr^{-n-2} [(\vec{r} . \vec{r})\vec{a}-(\vec{r}.\vec{a})\vec{r}] + r^{-n} (2\vec{a}) ##

    ## = -nr^{-n-2} r^2\vec{a} + nr^{-n-2}(\vec{r} . \vec{a})\vec{r} + 2r^{-n}\vec{a} ##

    phew..simplifying further will yield the RHS.

    Note that I've used the following:

    ## ∇r^n=nr^{n-2} \vec{r}##

    ##∇ × (\vec{a} × \vec{r}) = 2\vec{a} ##

    Hard way:

    You can do it by finding the actual cross product on the LHS like I did, keeping in mind that r is not a constant scalar xD

    Also, later on you have to make an adjustment by adding and subtracting ## \frac{n}{r^{n+2}} a_{1,2,3} x^2 ## in each component of the cross product.
     
    Last edited: Nov 19, 2013
  6. Nov 19, 2013 #5
    From my book

    I've made many corrections to my last post
     

    Attached Files:

  7. Nov 19, 2013 #6
    Thanks a lot MrWarlock! :)

    The alternative method is much better.
     
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