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!

Calculating flux via divergence theorem.

  1. Jun 7, 2013 #1
    1. The problem statement, all variables and given/known data
    Compute the flux of [itex] \vec{F} [/itex] through [itex]z=e^{1-r^2} [/itex] where [itex] \vec{F} = [x,y,2-2z]^T [/itex] and [itex] r=\sqrt{x^2+y^2} [/itex].

    EDIT: the curve must satisfy [itex] z\geq 0 [/itex].


    2. Relevant equations
    Divergence theorem: [tex] \iint\limits_{\partial X} \Phi_{\vec{F}} = \iiint\limits_X \nabla\cdot\vec{F}\,dx\,dy\,dz [/tex]


    3. The attempt at a solution

    For the given [itex] \vec{F} [/itex], we have [itex] \nabla\cdot\vec{F} = 0 [/itex]. So isn't the flux just zero by the divergence theorem? I am confused because there is a hint saying that I should change the given surface to a simpler one.
     
    Last edited: Jun 7, 2013
  2. jcsd
  3. Jun 7, 2013 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    hi dustbin! :wink:
    correct :smile:

    but that's only for a closed surface …

    so find another (simpler) surface that you can join to this surface to make a closed surface :wink:
     
  4. Jun 7, 2013 #3
    Thanks for the tip tiny-tim! I should note that I forgot to put the restriction [itex] z\geq 1 [/itex] on the given surface. I will think about your suggestion and post back!
     
  5. Jun 7, 2013 #4
    Since [itex] X [/itex] must be a compact domain in [itex] \mathbb{R}^3 [/itex], we must bound from below the region bounded above by the given surface. Since [itex] z\geq 1 [/itex], setting [itex] 1=e^{1-r^2} [/itex] gives the (simpler) surface [itex] x^2+y^2=1 [/itex]. The union of this disc and the given surface form the boundary, [itex] \partial X [/itex], of a compact region [itex] X [/itex] of [itex] \mathbb{R}^3 [/itex]. Hence we may now apply the divergence theorem.
     
  6. Jun 7, 2013 #5

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    hi dustbin! :smile:

    yes, i think that's right

    (except that i'm not sure which surface you mean … x2+y2 = 1 is a cylinder :wink:)
     
  7. Jun 7, 2013 #6
    I meant [itex] x^2+y^2 = 1 [/itex] to be confined to the plane [itex] z=1 [/itex]. Thank you! :redface:

    To take it all the way:

    Call [itex] S_1 [/itex] the given surface and [itex] S_2 [/itex] the new surface so that [itex] S_1\cup S_2 = \partial X [/itex]. Observe that [itex] \Phi_{\vec{F}} = x\,dy\wedge dz - y\,dx\wedge dz + (2-2z)\,dx\wedge dy [/itex]. Parametrize [itex] S_2 [/itex] via [itex] x=r\cos\theta, \ y=r\sin\theta, \ z=1 [/itex] such that [itex] \theta \in [0,2\pi) \ , \ 0\leq r \leq 1 [/itex]. Call this parametrization [itex] \gamma [/itex]. Then [itex] \Phi_{\vec{F}}(\gamma) = (2-2)r = 0. [/itex] Applying the theorem, we have

    [tex]
    0 = \iiint\limits_X \nabla\cdot\vec{F}\,dx\,dy\,dz = \iint\limits_{S_1} \Phi_{\vec{F}} + \iint\limits_{S_2} \Phi_{\vec{F}} = \iint\limits_{S_1} \Phi_{\vec{F}}.
    [/tex]

    Thus the flux of [itex] \vec{F} [/itex] across [itex] S_1 [/itex] is 0.
     
  8. Jun 7, 2013 #7

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    that's very complicated :redface:

    isn't it simpler to keep to x,y,z. and say that F on the surface is (x,y,0)T, and so … ? :wink:
     
  9. Jun 7, 2013 #8
    The way that I did it is the only way I know to calculate flux. :redface:
     
  10. Jun 7, 2013 #9

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    flux is just F·ñ

    why do you need to convert to polar coordinates to calculate what (on this surface) is obviously 0 ?? :wink:
     
  11. Jun 7, 2013 #10
    Is [itex] \hat{\textbf{n}} [/itex] the orienting normal?
     
  12. Jun 7, 2013 #11

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    yüp! :smile:
     
  13. Jun 7, 2013 #12
    I see. The text I use doesn't use standard notation, so when I look at other books or sites about vector calculus, I feel hopelessly lost with the notation!

    Thank you for your help :-)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Calculating flux via divergence theorem.
Loading...