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Sidana [21]
3 years ago
13

????⃗ (x,y)=(3x−4y)????⃗ +2x????⃗ F→(x,y)=(3x−4y)i→+2xj→ and ????C is the counter-clockwise oriented sector of a circle centered

at the origin with radius 44 and central angle ????/3π/3. Use Green's theorem to calculate the circulation of ????⃗ F→ around ????C.
Mathematics
1 answer:
Karolina [17]3 years ago
6 0

By Green's theorem, the integral of \vec F along C is

\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_D\left(\frac{\partial(2x)}{\partial x}-\frac{\partial(3x-4y)}{\partial y}\right)\,\mathrm dx\,\mathrm dy=6\iint_D\mathrm dx\,\mathrm dy

which is 6 times the area of D, the region with C as its boundary.

We can compute the integral by converting to polar coordinates, or simply recalling the formula for a circular sector from geometry: Given a sector with central angle \theta and radius r, the area A of the sector is proportional to the circle's overall area according to

\dfrac A{\frac\pi3\,\rm rad}=\dfrac{16\pi}{2\pi\,\rm rad}\implies A=\dfrac{8\pi}3

so that the value of the integral is

\dfrac{6\times8\pi}3=\boxed{16\pi}

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