Use Stokes' Theorem to evaluate
C
F · dr
where C is oriented counterclockwise as viewed from above.
F(x, y, z) = xyi + 5zj + 7yk,
C is the curve of intersection of the plane
x + z = 8
and the cylinder
x2 + y2 = 9.
To use Stokes' Theorem to evaluate the line integral, we need to find the curl of the vector field F and the orientation of the curve C. Let's go step by step to solve this problem.
Step 1: Find the curl of the vector field F.
The curl of a vector field F = P i + Q j + R k is given by ∇ × F, where ∇ is the del operator (∇ = ∂/∂x i + ∂/∂y j + ∂/∂z k).
In this case, F(x, y, z) = xyi + 5zj + 7yk, so we have:
P = xy, Q = 5z, R = 7
Now, let's calculate the curl:
∇ × F = (∂R/∂y - ∂Q/∂z)i + (∂P/∂z - ∂R/∂x)j + (∂Q/∂x - ∂P/∂y)k
∂R/∂y = 0
∂Q/∂z = 5
∂P/∂z = 0
∂R/∂x = 0
∂Q/∂x = 0
∂P/∂y = 0
Substituting these values, we get:
∇ × F = 0i + 5j + 0k
∇ × F = 5j
So, the curl of the vector field F is 5j.
Step 2: Determine the orientation and curve C.
C is the curve of intersection of the plane x + z = 8 and the cylinder x^2 + y^2 = 9.
Let's first find the parametric equations for C. Rearranging the equation x + z = 8, we have z = 8 - x.
Since C lies on the cylinder x^2 + y^2 = 9, we can express x and y in terms of a parameter θ as follows:
x = 3cos(θ)
y = 3sin(θ)
Substituting these expressions back into the equation z = 8 - x, we get:
z = 8 - 3cos(θ)
Combined, the parametric equations for C are:
r(θ) = 3cos(θ)i + 3sin(θ)j + (8 - 3cos(θ))k
The orientation of C is counterclockwise as viewed from above.
Step 3: Evaluate the line integral using Stokes' Theorem.
According to Stokes' Theorem:
∫∫(∇ × F) · dS = ∮F · dr
Here, we want to evaluate the line integral ∮F · dr, so we need to calculate the surface integral on the left-hand side of the equation.
Since the curve C is bounded by the cylinder x^2 + y^2 = 9, we can use the region D defined by -π ≤ θ ≤ π.
The normal vector to the surface S defined by C is given by:
n = (∇r × ∇θ)/‖∇r × ∇θ‖
= (-3cos(θ)i - 3sin(θ)j + 3cos(θ)k) × (-3sin(θ)i + 3cos(θ)j)
= -9cos(θ)k
Now, let's evaluate the surface integral:
∫∫(∇ × F) · dS = ∫∫(-9cos(θ)k) · (9cos(θ)dA)
Since k · k = 1, the integral simplifies to:
∫∫(-9cos(θ)k) · (9cos(θ)dA) = -81∫∫cos^2(θ)dA
To evaluate this integral over the region D, we can use the fact that cos^2(θ) = (1 + cos(2θ))/2.
Substituting this into the integral, we get:
-81∫∫[(1 + cos(2θ))/2]dA
Now, we can perform the double integral over the region D using suitable techniques such as polar coordinates or parametric equations.
Once we have the result of the surface integral, it will give us the value of the line integral ∮F · dr.