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Laplace equation in spherical coordinates

  1. Nov 9, 2011 #1
    1. The problem statement, all variables and given/known data
    Verify by direct substitution in Laplace's equation that the functions (2.19) are harmonic in in appropriate domains in ℝ2


    2. Relevant equations
    (2.19)= [tex]{u_n(r, \theta)= \lbrace{1,r^{n}cos(n \theta), r^{n}sin(n \theta), n= 1, 2...; log(r), r^{-n}cos(n \theta), r^{-n}sin(n \theta); n= 1, 2,...; \rbrace}}[/tex]

    Should look like a piece-wise function (Don't know how to do that in latex).

    Laplace equation in spherical coordinates
    [tex]{\Delta u = \frac{1}{r^{n-1}} \frac{\partial (r^{n-1}\frac{\partial u(r, \theta)}{\partial r})}{\partial r} + \frac{1}{r^{2}} \frac{\partial^2 u(r, \theta)}{\partial^2 \theta} = 0}[/tex]


    3. The attempt at a solution

    I don't what I'm suppose to substitute, do I substitute the whole thing into the Laplace equation, each time I see u(r, θ)?
     
  2. jcsd
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