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27 February, 07:32

Suppose F⃗ (x, y) = 4yi⃗ + 2xyj⃗. Use Green's Theorem to calculate the circulation of F⃗ around the perimeter of a circle C of radius 3 centered at the origin and oriented counter-clockwise.

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  1. 27 February, 07:52
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    From Green's theorem, the circulation of a function F (x, y) around a circle is given as

    ∫ (F (x, y). dA = Area of the circle

    π (3^2) - π (0^2) = 9π

    Since the result is oriented counter-clockwise, the result will take negative value.

    The circulation of F (x, y) is - 9π

    Step-by-step explanation:

    ∫c (4y dx + 2xy dy)

    = ∫∫ [ (∂/∂x) (2xy) - (∂/∂y) (4y) ] dA, by Green's Theorem

    By integrating the function F (x, y) = 4yi + 2xyj, around the circle, the result is πr2[3, 0], from origin 0, to radius of 3
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