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.

Answer 1

Answer:

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

Answer 2

Answer:

-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