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Scale Factors and Jacobians

  1. Mar 12, 2012 #1
    So I've been trying to figure out how to find the surface area Jacobians in spherical coordinates (I know how to use it to find the volume Jacobian). Using the divergence theorem I was able to find these Jacobians top-down, however, I am unsure to how one would derive them in the first place. I have tried playing around with matrix that maps your r, θ, [itex]\phi[/itex] onto x, y, z, without success. I know that there is something to do with a tangent plane here as well, but this is where I get very lost.

    Scouring the interwebs, I found that the scale factors matched my surface area Jacobians (well not directly, but they matched what I needed to multiply into my surface integral when I was using d[itex]\phi[/itex] and d[itex]\theta[/itex] to "sweep" across my surface. To explain this better I will list the scale factors:
    h_r=1
    h_[itex]\theta[/itex]=rsin[itex]\phi[/itex]
    h_[itex]\phi[/itex]=r
    So if I was trying to find the area of a face with normal vector r I would need to use h_[itex]\theta[/itex]=rsin[itex]\phi[/itex] and h_[itex]\phi[/itex]=r (since d[itex]\phi and d\theta[/itex] sweep across this face), so r^2sin[itex]\phi[/itex], times d[itex]\theta[/itex]d[itex]\phi[/itex] to find the area of this face.)

    My question now is how do I find these scales factors in a non-top-down manner. I tried looking here: http://mathworld.wolfram.com/ScaleFactor.html, but it was in Chinese.

    I am teaching myself this so please correct me if I am using the wrong terms (in particular, I doubt there are such things called a surface area or volume Jacobian, I just lack a better, more correct descriptor).
     
  2. jcsd
  3. Mar 12, 2012 #2

    arildno

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    What, exactly, is a "non-top-down" manner??
     
  4. Mar 12, 2012 #3

    mathwonk

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    what do spherical coordinates have to do with surface area?
     
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