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Green's Theorem #3
Suppose that $\text{div}\overrightarrow{F} = 2x + 3y$. Find the approximate flux of a vector field across a circle of
radius 0.1 centered at the point (1, 1).
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So basically we are given $$\int_C F\cdot dn=\int \int divF=\int \int 2x+3y$$ It is a lot easier to do this in polar coordinates so we have $x=0.1cos(\theta)$ and $y=0.1sin(\theta)$ thus $divF=0.2cos(\theta)+0.3sin(\theta)$. This just becomes a matter of integration so:
$\int_C F\cdot dn= \int_{0}^{2\pi} \int_{0}^{1} 0.2cos(\theta)+0.3sin(\theta) \space drd\theta=$
$\int_{0}^{2\pi} 0.2cos(\theta)+0.3sin(\theta) \space d\theta=0.2 sin(\theta)-0.3cos(\theta)=0$