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Math Help - Converting Coordinate Systems

  1. #1
    Super Member craig's Avatar
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    Converting Coordinate Systems

    Bit stuck on converting from Cartesian to Spherical:

    Let S be the spherical surface defined by x^2 + y^2 + z^2 = a^2, and let \mathbf{F} = z^3 \mathbf{k}.

    Using SPs (r,\theta,\phi), and with \mathbf{\delta S} = a^2 \sin{\theta} \delta \theta \delta \phi \mathbf{\hat{r}}

    i) Show \mathbf{F} \cdot \mathbf{\delta S} = a^5 \cos^4{\theta} \sin{\theta} \delta \theta \delta \phi

    ii) Calculate \int_S \mathbf{F} \cdot \mathbf{dS}.

    i)Well converting \mathbf{F} to Spherical, we get (z^3,0,0), so the scalar product gives:

    z^3(a^2 \sin{\theta} \delta \theta \delta \phi).

    Now we also know that z = a \cos{\theta}, either from using z = a^2 - (x^2 + y^2), or from the definition of SPs.

    So substituting this in gives:

    \cos^3{\theta} a^3(a^2 \sin{\theta} \delta \theta \delta \phi)

    Which simplifies to:

    a^5 \cos^3{\theta} \sin{\theta} \delta \theta \delta \phi

    I'm just at a bit of a loss where they get their extra \cos{\theta} from?

    ii) Integrating this gives us a^5 \int^{2 \pi}_0 d\phi \int^{pi}_0 \cos^4{\theta} \sin{\theta} d\theta

    \frac{-2 \pi a^5}{5} \left[\cos^5{\theta} \right]^{\pi}_0

    Which gives \frac{4 \pi a^5}{5}, would this be right?

    Thanks in advance for any help
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  2. #2
    Super Member craig's Avatar
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    Would the first bit involving the extra \cos{\theta} be because:

    \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos{\theta}?

    I think this would explain where they've got the extra \cos{\theta} from?
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