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Math Help - Stoke's Theorem

  1. #1
    s7b
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    Stoke's Theorem

    Use the surface integral in Stoke's Theorem to calculate the circulation of the field F around the curve C in the indicated direction: F=x^2y^3i + j + zk, C: The intersection of the cylinder x^2+y^2=4 and the hemisphere x^2+y^2+z^2=16 , z greater or equal to 0, counter clockwise when viewed from above.

    Can someone please help! I'm lost when it comes to Stokes!
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    Quote Originally Posted by s7b View Post
    Use the surface integral in Stoke's Theorem to calculate the circulation of the field F around the curve C in the indicated direction: F=x^2y^3i + j + zk, C: The intersection of the cylinder x^2+y^2=4 and the hemisphere x^2+y^2+z^2=16 , z greater or equal to 0, counter clockwise when viewed from above.

    Can someone please help! I'm lost when it comes to Stokes!
    first we see that \text{curl}(\bold{F}) = -3x^2y^2 \bold{k}. we have S: \ z=2\sqrt{3} and D=\{(x,y): \ x^2+y^2 \leq 4 \}. clealy, the upward normal unit of S is \bold{n}=\bold{k}. also we have dS=dxdy. therefore:

    \int \int_D \text{curl}(\bold{F}) \cdot \bold{n} \ dS=-3 \int \int_{x^2 +y^2 \leq 4} x^2y^2 \ dxdy=-8 \pi.
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