Question

Use Green's Theorem to calculate the circulation of G around the curve, oriented counterclockwise. G = 7yi + xyj around the circle of radius 6 centered at the origin. Close int C G. dr= [ ]

Answer 1

The circle's perimeter is determined by F = -9π, according to green's theorem.

What is Green's theorem?

The simple formula for Green's theorem is the link between the total amount of microscopic circulation inside curve C and the macroscopic circulation revolving around it.

Remembering that we are talking about two dimensions, if C is a straightforward closed curve in the plane, it will enclose the region D (shown in red) in the plane.

So, knowing that this exercise should be solved using Green's theorem, then:

$\begin{aligned}& \int 4 y d x+2 x y d y=\iint(d / d x)(2 x y)-(d / d y)(4 y) d A \\& =\iint 2 y-4 d A\end{aligned}$

You must convert in this manner to polar coordinates, and the result will be:

$\begin{aligned}& \int_0^3 \int_0^{2 \pi}(2 r \sin (\theta)-4)(r) d \theta d r \\& =\int_0^3(-2 r \cos (\theta)-4 \theta)(-2 r \pi) d r \\& =-(\pi)\left(r^2\right)=-9 \pi\end{aligned}$

Therefore, the circle's perimeter is determined by F = -9π, according to green's theorem.

Know more about Green's theorem here:

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Correct 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.