Suppose F⃗(x,y)=⟨2y,−sin(y)⟩ and C is the circle of radius 9 centered at the origin oriented counterclockwise. Using your parametrization in part (a), set up an integral for calculating the circulation of F⃗ around C.
the circulation is ∫F(r) dr = area * ∇xF
you can choose which makes you more comfortable.
To calculate the circulation of F⃗ around C, we need to evaluate the line integral of the dot product of F⃗ and dR⃗ along the curve C.
Let's first find the parametrization of C in terms of t.
Circumference of a circle, C = 2πr, where r is the radius. In this case, the radius of the circle is 9, so the circumference is 2π(9) = 18π.
We can use the parametric equations for a circle centered at the origin:
x = r * cos(t)
y = r * sin(t)
In this case, r = 9, so the parametric equations become:
x = 9 * cos(t)
y = 9 * sin(t)
Now, let's find the differential of the position vector dR⃗:
dR⃗ = ⟨dx, dy⟩ = ⟨dx/dt, dy/dt⟩ * dt
Since x = 9 * cos(t) and y = 9 * sin(t), we can differentiate these equations with respect to t:
dx/dt = -9 * sin(t)
dy/dt = 9 * cos(t)
Therefore, dR⃗ = ⟨-9 * sin(t), 9 * cos(t)⟩ * dt.
Next, let's calculate the dot product of F⃗ and dR⃗:
F⃗ · dR⃗ = ⟨2y, -sin(y)⟩ · ⟨-9 * sin(t), 9 * cos(t)⟩ * dt
= (2y * -9 * sin(t)) + (-sin(y) * 9 * cos(t)) * dt
= -18y * sin(t) - 9 * sin(y) * cos(t) * dt
Now, to set up the integral for calculating the circulation of F⃗ around C, we integrate the dot product along the curve C:
∮ F⃗ · dR⃗ = ∫C (-18y * sin(t) - 9 * sin(y) * cos(t)) * dt
So the integral for calculating the circulation of F⃗ around C is:
∮ F⃗ · dR⃗ = ∫C (-18y * sin(t) - 9 * sin(y) * cos(t)) * dt
To set up an integral for calculating the circulation of a vector field F⃗ around a curve C, we can use the line integral formula:
Circulation = ∮ F⃗ · dr⃗
where F⃗ is the vector field, dr⃗ is the differential displacement vector along the curve C, and ∮ represents the line integral around the curve.
In this case, we have the vector field F⃗(x, y) = ⟨2y, −sin(y)⟩ and the curve C is a circle of radius 9 centered at the origin, oriented counterclockwise.
To parametrize the curve C, we can use polar coordinates. Let's denote the angle parameter as θ, which varies from 0 to 2π as we go around the circle once. The radius of the circle is 9, so the position vector r⃗ can be expressed as:
r⃗(θ) = ⟨9cosθ, 9sinθ⟩
Now, let's find the differential displacement vector dr⃗:
dr⃗ = ⟨dx, dy⟩
We can find dx and dy by taking the derivatives of x and y with respect to θ:
dx = (dr⃗/dθ) · (dθ/dx) = (-9sinθ) · (1/(-9sinθ)) = -dθ
dy = (dr⃗/dθ) · (dθ/dy) = (9cosθ) · (1/(9cosθ)) = dθ
So, dr⃗ = ⟨-dθ, dθ⟩
Now, substitute the vector field F⃗ and the differential displacement vector dr⃗ into the circulation formula:
Circulation = ∮ F⃗ · dr⃗
= ∮ ⟨2y, -sin(y)⟩ · ⟨-dθ, dθ⟩
= ∮ (2y)(-dθ) + (-sin(y))(dθ)
= ∮ -2ydθ - sin(y)dθ
To evaluate this integral, we need to express y in terms of θ. Since y is the y-coordinate of the position vector r⃗, we have y = 9sinθ.
Substituting this into the integral, we get:
Circulation = ∮ -2(9sinθ)dθ - sin(9sinθ)dθ
Now, you can proceed with evaluating the integral over the appropriate range of the angle parameter θ (from 0 to 2π) to calculate the circulation of F⃗ around C.