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Vector field flow over surface in 3D

  1. Jan 7, 2014 #1
    1. The problem statement, all variables and given/known data
    Calculate the flow of ##\vec{F}=(y^2,x^2,x^2y^2)## over surface ##S## defined as ##x^2+y^2+z^2=R^2## for ##z \geq 0## with normal pointed away from the origin.


    2. Relevant equations



    3. The attempt at a solution

    The easiest was is probably with Gaussian law. I would be really happy if somebody could correct me if I am wrong and answer my question below:

    Gaussian law: ##\int \int _O\vec{F}d\vec{S}+\int \int _S\vec{F}d\vec{S}=\int \int \int_{Body} \nabla\vec{F}dV## where I used notation ##O## for the circle.

    Now ##\nabla\vec{F}= 0## therefore ##\int \int _O\vec{F}d\vec{S}+\int \int _S\vec{F}d\vec{S}=0## so all that remains is to calculate the floe through surface ##O##.

    Using polar coordinates ##x=r \cos \varphi ## and ##y= r \sin \varphi## for ##z=0##. Than ##r_{\varphi } \times r_{r}=(0,0,-r)##

    ##\int \int _O\vec{F}d\vec{S}=-\int_{0}^{2\pi }\int_{0}^{R}r^{5} \cos^2 \varphi \sin^2 \varphi d\varphi dr##

    That should be ##-\frac{\pi R^6}{96}##.

    Question here: I am a bit confused weather I should use the other sign here ##r_{\varphi } \times r_{r}=(0,0,-r)## or is this the right one?
     
  2. jcsd
  3. Jan 7, 2014 #2

    vanhees71

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    Science Advisor
    2016 Award

    Gauß's Law is for all the surface normal vectors pointing away from the enclosed volume. So your idea is correct.
     
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