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Tensor and series questions

  1. Aug 27, 2016 #1
    1. The problem statement, all variables and given/known data
    i have a few homework question and want to be sure if I have solved them right.
    Q1) Write ##\vec{\triangledown}\cdot\vec{\triangledown}\times\vec{A}## and ##\vec{\triangledown}\times\vec{\triangledown}\phi## in tensor index notation in ##R^3##

    Q2) the spherical coordinates
    ##x=r sin\theta cos\phi##
    ##y=r sin\theta sin\phi##
    ##z=r cos\theta##
    what is the relataion of ##dx, dy, dz## in terms of ## dr , d\theta , d\phi , ##

    Q3) Determine whether the following series converges
    ##\sum \left(\frac{2}{5^{k+1} }+\frac{(2k)!}{3^k}\right)##
    this problem has no summation startpoint. I thought such question must have a start point and go to infinity. like k=0 or k=2


    3. The attempt at a solution
    Q1) ##\vec{\triangledown}\cdot\vec{\triangledown}\times\vec{A} = \epsilon_{ijk}\partial_{i}\partial_{j}A_{k}##
    ##\vec{\triangledown}\times\vec{\triangledown}\phi = \epsilon_{ijk}\partial_{j}\partial_{k}\phi##

    Q2) ##dx=sin\theta cos\phi dr + r cos\theta cos\phi d\theta - r sin\theta sin\phi dphi##
    ##dy=sin\theta sin\phi dr + r cos\theta sin\phi d\phi + r cos\theta cos\phi d\phi##
    ## dz=cos\theta dr - r sin\theta d\theta##

    Q3)
    ##\sum \frac{2}{5^{k+1} }=\frac{2}{5}+\frac{2}{25}+\frac{2}{125}+....##
    from geometric series
    ##\lim_{k \rightarrow \infty}S_n=\frac{a}{1-r}##

    ##a=\frac{2}{5}## ##r=\frac{1}{5}##

    since ##\mid r\mid<1## the series converges
    ##\lim_{k \rightarrow \infty}S_n=\frac{\frac{2}{5}}{1-\frac{1}{5}}=\frac{1}{2}##

    ##\sum \frac{(2k)!}{3^k}##

    using D'Alembert ratio test
    ##\rho=\lim_{k \rightarrow \infty} \frac{u_{k+1}}{u_{k}}##

    ##\rho=\lim_{k \rightarrow \infty} \frac{\frac{[2(k+1)]!}{3^{k+1}}}{\frac{(2k)!}{3^{k}}}##


    ##\rho=\lim_{k \rightarrow \infty} \frac{(2k+2)!3^{k}}{(2k)!3^{k+1}}##


    ##\rho=\lim_{k \rightarrow \infty} \frac{(2k+2)(2k+1)}{3}##

    this gives infinity and therefore this series diverges
     
  2. jcsd
  3. Aug 27, 2016 #2

    Orodruin

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    Q1) The first expression is correct. The second is missing a basis vector and has a mismatched free index on the right hand side.

    Q2) Double check the last term of dy.

    Q3) ok
     
  4. Aug 28, 2016 #3
    Q1) the second expression. I think ##\phi## is a scalar and therefore it should not have an index. Right?

    Q2) yes i made a mistak. It should be
    ##dy=sin\theta sin\phi dr + r cos\theta sin\phi d\theta + r sin\theta cos\phi d\phi##

    Q3) so the final answer is
    Converge + Diverge = Diverge.
    And is it normal that the Summation has no startpoint? I thought there is a trick.
     
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