1. Not finding help here? Sign up for a free 30min 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!

Calculate flux with normal form of Green's theorem

  1. Jun 11, 2013 #1
    1. The problem statement, all variables and given/known data

    Let [itex] R [/itex] be the region bounded by the lines [itex] y=1 [/itex], [itex] y=0 [/itex], [itex] xy=1 [/itex], and [itex] x=2 [/itex]. Let [itex] \vec{F} = \begin{bmatrix} x^4 & y^2-4x^3y \end{bmatrix}^T [/itex]. Calculate the outward flux of [itex] \vec{F} [/itex] over the boundary of [itex] R [/itex].

    2. Relevant equations

    Green's theorem (normal form): [itex] \int_{\partial R} F_1\,dy - F_2\,dx = \iint_R F_{1_x} + F_{2_y}\,dx\,dy [/itex].

    3. The attempt at a solution

    We have [itex] F_{1_x}+F_{2_y} = 4x^3+2y-4x^3 = 2y [/itex]. Then
    [tex]
    \begin{align*}
    \iint_R 2y\,dx\,dy &= \int_0^1\int_0^x 2y\,dy\,dx + \int_1^2\int_0^{\frac{1}{x}} 2y\,dy\,dx \\
    &= \int_0^1 x^2\,dx + \int_1^2 \frac{1}{x^2}\,dx \\
    &= \frac{1}{3} + \frac{1}{2} = \frac{5}{6} \ .
    \end{align*}
    [/tex]
    By Green's theorem, the flux is [itex] \frac{5}{6} [/itex].
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?



Similar Discussions: Calculate flux with normal form of Green's theorem
Loading...