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Curvilinear Coordinates and Vector Calculus

  1. Nov 4, 2011 #1
    1. The problem statement, all variables and given/known data

    With [itex]\vec{L}[/itex] = -i[itex]\vec{r}[/itex] x [itex]\nabla[/itex], verify the operator identities

    [itex]\nabla = \hat{r}\frac{\partial }{\partial \vec{r}}-i\frac{\vec{r}\times\vec{L}}{r^{2}}[/itex]
    [itex]\vec{r} \bigtriangledown ^2 - \bigtriangledown (1+\vec{r}\frac{\partial }{\partial \vec{r}})=i\bigtriangledown \times \vec{L}[/itex]

    2. Relevant equations

    3. The attempt at a solution

    ... I tried to expand it, and use some identities... But the equation becomes super complicated... Help!
  2. jcsd
  3. Nov 4, 2011 #2


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    I don't see other way to do it other than expanding and writing the nabla operator in spherical coordinates...The only identity you need is the one for double vector product

    [tex] \vec{A}\times \left(\vec{B}\times\vec{C}\right) = \ ... [/tex]
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