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Homework Help: 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
    \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} \ .
    By Green's theorem, the flux is [itex] \frac{5}{6} [/itex].
  2. jcsd
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