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Math Help - Divergence Theorem

  1. #1
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    Divergence Theorem

    The question wants me to use Gauss's theorem
    \vec{F} = x\hat{i} + y\hat{j} + {z}^{2}\hat{k}
    Evaluate teh flux on the closed surface it is a cylinder:
    {x}^{2} + {y}^{2} \leq {h}^{2}
    whereby
    0 \leq z \leq b

    ---- attempt ----
    Flux = \iiint \nabla . \vec{F} dV
    Flux = \iiint (2 + 2Z) dV

    Now i could say dV = dx dy dz but i do not have any arbitary values for x and y do i convert to cylindrical polar coordiantes and use r is between r and 0 and the angle is between 2pi and zero?

    Thanks!
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by imagemania View Post
    The question wants me to use Gauss's theorem
    \vec{F} = x\hat{i} + y\hat{j} + {z}^{2}\hat{k}
    Evaluate teh flux on the closed surface it is a cylinder:
    {x}^{2} + {y}^{2} \leq {h}^{2}
    whereby
    0 \leq z \leq b

    ---- attempt ----
    Flux = \iiint \nabla . \vec{F} dV
    Flux = \iiint (2 + 2Z) dV

    Now i could say dV = dx dy dz but i do not have any arbitary values for x and y do i convert to cylindrical polar coordiantes and use r is between r and 0 and the angle is between 2pi and zero?

    Thanks!
    Your divergence is correct. The divergence of F

    \nabla \cdot \mathbf{F}=1+1+2z

    The radius of the cylinder is h and z goes from zero to b and your angle is correct. Try to evaluate the integral.
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  3. #3
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    Ok sure, going from my original idea:
    \int^{b}_{0}\int^{2\pi}_{0}\int^{h}_{0} (2+2z) drd\phi dz
    Hence i obtain:
    2\pi hb(2+b)

    Is this satisfactory?
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  4. #4
    Behold, the power of SARDINES!
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    yes that is the correct solution
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