## 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 centere

Question

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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6 days 2021-10-08T00:41:25+00:00 2 Answers 0

-9π

Step-by-step explanation:

∫c (4y dx + 2xy dy)

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

= ∫∫ (2y – 4) dA

Now convert to polar coordinates:

∫(r = 0 to 3) ∫(θ = 0 to 2π) (2r sin θ – 4) * (r dθ dr) — first integration

= ∫(r = 0 to 3) (-2r cos θ – 4θ) * r {for θ = 0 to 2π} dr

= ∫(r = 0 to 3) -2πr dr

= -πr² {for r = 0 to 3}

= -π(3²) – -π(0)²

= -9π

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