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 evaluate the line integral using Stokes' Theorem, we need to follow these steps:
Step 1: Find the curl of the vector field F.
Step 2: Determine the surface S that is bounded by the curve of intersection C.
Step 3: Calculate the surface integral of the curl of F over S.
Step 1: Finding the curl of F:
The curl of a vector field F = P i + Q j + R k is given by the determinant:
curl(F) = (dR/dy - dQ/dz) i + (dP/dz - dR/dx) j + (dQ/dx - dP/dy) k
In our case, F(x, y, z) = xy i + 5z j + 7y k.
So, P = xy, Q = 0, and R = 5z.
Now, let's calculate each partial derivative:
dR/dy = 0
dQ/dz = 0
dP/dz = 0
dR/dx = 0
dQ/dx = 0
dP/dy = x
Therefore, the curl of F is (0 - 0)i + (0 - 0)j + (x - 0)k = xk.
Step 2: Determining the surface S:
The curve of intersection C is defined by the plane x + z = 8 and the cylinder x^2 + y^2 = 9.
To parametrize the curve of intersection C, we can solve these two equations simultaneously:
x + z = 8 => z = 8 - x [Equation 1]
x^2 + y^2 = 9 [Equation 2]
By substituting Equation 1 into Equation 2, we get:
x^2 + y^2 = 9 => x^2 + y^2 = 9 - x^2 => 2x^2 + y^2 = 9.
Let's rewrite Equation 2 in terms of y:
y = ±√(9 - 2x^2).
Now we can parametrize the curve of intersection C:
r(x, y) = x i + y j + (8 - x) k [Parametrization of C]
Step 3: Calculating the surface integral:
By using Stokes' Theorem, we have:
∬_S curl(F) · dS = ∮_C F · dr
We've already found that curl(F) = xk and a parametrization for C is r(x, y) = x i + y j + (8 - x) k. Thus, we can rewrite the line integral as:
∮_C F · dr = ∮_C (xy i + 5z j + 7y k) · (dx i + dy j + dz k)
= ∮_C (xy dx + 5z dy + 7y dz)
Now, we need to convert the line integral into a double integral over the region D in the xy plane.
To do that, we integrate the above expression with respect to x and y from the boundaries of D.
Let's solve it step by step, using Green's Theorem (a special case of Stokes' Theorem):
∮_C (xy dx + 5z dy + 7y dz) = ∬_D (∂(7y)/∂x - ∂(xy)/∂y) dA
The boundaries of D are determined by the cylinder x^2 + y^2 = 9. So, let's rewrite our expression:
∬_D (∂(7y)/∂x - ∂(xy)/∂y) dA = ∫_(θ=0 to 2π) ∫_(r=0 to 3) ((7)(r sinθ) - (r cosθ)(r)) r dr dθ
Now, evaluate this double integral to get the final result.