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Math Help - surface integrals

  1. #1
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    surface integrals

    evaluate double int over S (e^z) dS
    where S is the surface of the sphere x^2 + y^2 + z^2 = a^2

    I know dS = a^2(sintheta)(d.theta)(d.phi) using the parametrization

    r = (a.sintheta.cosphi,a.sintheta.sinphi,acosphi)

    so from there do I just calculate

    int int e^(acosphi) a^2(sintheta)(d.theta)(d.phi)

    ?

    many thanks
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  2. #2
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    Quote Originally Posted by James0502 View Post
    evaluate double int over S (e^z) dS
    where S is the surface of the sphere x^2 + y^2 + z^2 = a^2

    I know dS = a^2(sintheta)(d.theta)(d.phi) using the parametrization

    r = (a.sintheta.cosphi,a.sintheta.sinphi,acosphi)

    so from there do I just calculate

    int int e^(acosphi) a^2(sintheta)(d.theta)(d.phi)

    ?

    many thanks
    The surface can be parametrized as \bold{g} : [0,2\pi]\times [0,\pi] \to \mathbb{R}^3 as \bold{g}(\theta,\phi) = (a^2\cos \theta \sin \phi, a^2 \sin \theta \sin \phi, a^2 \cos \phi).

    The function that you have is f(x,y,z) = e^z.

    Therefore, \iint_S f dS = \int_0^{\pi} \int_0^{2\pi} f(\bold{g}(\theta,\phi)) \cdot \left| \frac{\partial \bold{g}}{\partial \theta} \times \frac{\partial \bold{g}}{\partial \phi} \right| d\theta ~ d\phi

    Compute that for your surface integral.
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