Mathematics, 17.03.2022 17:20 emmaogle2003
Find the circulation done by the velocity field V(x, y)= -y/[(x+1)^2 +4y^2]i + (x+1)/[(x+1)^2 +4y^2]j around the positively oriented square with vertices (2, 0), (0, 2), (−2, 0), and (0, −2).
Answers: 3
![\displaystyle \int_C \vec V(x,y) \cdot d\vec r = \iint_{\mathrm{int}(C)} \frac{\partial}{\partial x}\left[\frac{x+1}{(x+1)^2+4y^2}\right] - \frac{\partial}{\partial y}\left[-\frac{y}{(x+1)^2+4y^2}\right] \, dx\,dy](/tpl/images/2698/2674/d1940.png)
![\dfrac{\partial}{\partial x}\left[\dfrac{x+1}{(x+1)^2+4y^2}\right] = \dfrac{4y^2 - (x+1)^2}{((x+1)^2+4y^2)^2}](/tpl/images/2698/2674/ac846.png)
![\dfrac{\partial}{\partial y}\left[-\dfrac{y}{(x+1)^2+4y^2}\right] = \dfrac{4y^2-(x+1)^2}{((x+1)^2+4y^2)^2}](/tpl/images/2698/2674/a302f.png)
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Find the circulation done by the velocity field V(x, y)= -y/[(x+1)^2 +4y^2]i + (x+1)/[(x+1)^2 +4y^2]...
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