1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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

    User Avatar
    Science Advisor
    Gold Member
    2017 Award

    Gauß's Law is for all the surface normal vectors pointing away from the enclosed volume. So your idea is correct.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Loading...