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Find the circulation of F(x,y) = \langle 2y, -\sin y \rangle around the circle C of radius...

Question:

Find the circulation of {eq}\mathbf F(x,y) = \langle 2y, -\sin y \rangle {/eq} around the circle {eq}C {/eq} of radius 3 centered at the origin oriented counter clockwise.

Calculating the Circulation:

The objective is to calculate the circulation by using Green's theorem.

The general form of Green's theorem is {eq}\displaystyle \int _{C} F \cdot dr = \iint_{R} \left ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right ) dA {/eq}

By converting into polar coordinates, we have to find the limit for integration.

We have to integrate the function in the order of {eq}\displaystyle dr \ and \ \displaystyle d\theta {/eq}.

Answer and Explanation:

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The given vector function is:

{eq}\displaystyle F= 2y, -\sin y {/eq}

The given radius is {eq}3. {/eq}

By using polar coordinates,

{eq}x = r\cos...

See full answer below.


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Work as an Integral

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Chapter 7 / Lesson 9
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After watching this video, you will be able to solve calculus problems involving work and explain how that relates to the area under a force-displacement graph. A short quiz will follow.


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