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Math Help - surface integral help!

  1. #1
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    Question surface integral help!

    Evaluate the surface integral the integral of S of (x+y+z)ds, where S is the hemisphere x^2+y^2+z^2 = a^2 at z> or = 0.

    Thank you very much.
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  2. #2
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    Quote Originally Posted by kittycat View Post
    Evaluate the surface integral the integral of S of (x+y+z)ds, where S is the hemisphere x^2+y^2+z^2 = a^2 at z> or = 0.

    Thank you very much.
    We can parametrize the surface, (r,\theta)\mapsto (r\cos \theta, r\sin\theta, \sqrt{a^2-r^2}) where 0\leq \theta \leq 2\pi and 0\leq r\leq a, let us call that parametrization by \bold{G}(r,\theta) and the function (x,y,z)\mapsto x+y+z as f(x,y,z). That means the surface integral is,
    \iint_R f\left( \bold{G}(r,\theta) \right) \cdot \left| \frac{\partial \bold{G}}{\partial r}\times \frac{\partial \bold{G}}{\partial \theta} \right| dA where R is the rectangle 0\leq \theta \leq 2\pi and 0\leq r\leq a. Can you finish that?
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