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meriva
3 years ago
13

Use Green's Theorem to calculate the circulation of F =2xyi around the rectangle 0≤x≤8, 0≤y≤3, oriented counterclockwise.

Mathematics
1 answer:
Tamiku [17]3 years ago
6 0

Green's theorem says the circulation of \vec F along the rectangle's border C is equal to the integral of the curl of \vec F over the rectangle's interior D.

Given \vec F(x,y)=2xy\,\vec\imath, its curl is the determinant

\det\begin{bmatrix}\frac\partial{\partial x}&\frac\partial{\partial y}\\2xy&0\end{bmatrix}=\dfrac{\partial(0)}{\partial x}-\dfrac{\partial(2xy)}{\partial y}=-2x

So we have

\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_D-2x\,\mathrm dx\,\mathrm dy=-2\int_0^3\int_0^8x\,\mathrm dx\,\mathrm dy=\boxed{-192}

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