Use Stokes' Theorem to evaluate S
curl F · dS.
F(x, y, z) = x2z2i + y2z2j + xyzk,
S is the part of the paraboloid
z = x2 + y2
that lies inside the cylinder
x2 + y2 = 25,
oriented upward.
To use Stokes' Theorem to evaluate the surface integral of curl F dot dS, we need to follow these steps:
Step 1: Find the curl of F.
Step 2: Determine the orientation of the surface S.
Step 3: Compute the outward unit normal vector of S.
Step 4: Parameterize the surface S.
Step 5: Calculate the cross product of the tangent vectors of the parameterization.
Step 6: Compute the dot product of the curl of F and the cross product.
Step 7: Evaluate the resulting surface integral.
Let's go through these steps one by one.
Step 1: Find the curl of F:
To find the curl of F, we take the determinant of the matrix:
curl F =
| i j k |
| ∂/∂x ∂/∂y ∂/∂z |
| x^2z^2 y^2z^2 xy |
Differentiating each component of F with respect to x, y, and z, we get:
curl F = <2xyz^2, -2xyz^2, 2x^2yz - 2y^2z^2>
Step 2: Determine the orientation of the surface S:
The question states that the surface is oriented upward. This means that the surface normal vector should point outward.
Step 3: Compute the outward unit normal vector of S:
The normal vector for the surface z = x^2 + y^2 can be calculated by taking the gradient of the function:
∇g = <∂g/∂x, ∂g/∂y, -1>
= <2x, 2y, -1>
To normalize the vector, divide it by its magnitude:
N = <2x, 2y, -1> / √(4x^2 + 4y^2 + 1)
Since the surface is oriented upward, the outward unit normal vector becomes N = <-2x, -2y, 1> / √(4x^2 + 4y^2 + 1)
Step 4: Parameterize the surface S:
To parameterize the surface S, we can use the cylindrical coordinates (r, θ, z). Since the surface lies inside the cylinder x^2 + y^2 = 25, we have:
r = 5,
θ ∈ [0, 2π],
z = r^2 = 25.
So, the parameterization becomes:
r(θ) = <5cos(θ), 5sin(θ), 25>
Step 5: Calculate the cross product of the tangent vectors of the parameterization:
To find the cross product of the tangent vectors, we need to take the partial derivatives of the parameterization:
∂r/∂θ = <-5sin(θ), 5cos(θ), 0>
Now, taking the cross product:
∂r/∂θ x N = <-5sin(θ), 5cos(θ), 0> x <-2(5cos(θ)), -2(5sin(θ)), 1>
= <0, 0, -10sin^2(θ) - 10cos^2(θ)>
= <0, 0, -10>
Step 6: Compute the dot product of the curl of F and the cross product:
Taking the dot product:
curl F · (∂r/∂θ x N) = <2xyz^2, -2xyz^2, 2x^2yz - 2y^2z^2> · <0, 0, -10>
= 0 + 0 + (-20x^2yz - 20y^2z^2)
= -20(x^2yz + y^2z^2)
Step 7: Evaluate the resulting surface integral:
To evaluate the surface integral, we need to find the area of the surface S. Since S is the part of the paraboloid z = x^2 + y^2 inside the cylinder x^2 + y^2 = 25, we can calculate its surface area using a surface integral formula.
The surface area integral becomes:
∬S dS = ∫∫S ||∂r/∂θ x N|| dA,
where ||∂r/∂θ x N|| is the magnitude of the cross product of the tangent vectors, and dA is the area element.
Using the parameterization r(θ) = <5cos(θ), 5sin(θ), 25>, we can compute this integral:
∫∫S ||∂r/∂θ x N|| dA = ∫∫S 10 dA,
Since S is a circular surface, we can convert the integral to polar coordinates:
∫∫S 10 dA = ∫[0,2π]∫[0,5] 10 * r dr dθ.
Evaluating this integral gives us:
∫∫S 10 dA = 10 ∫[0,2π]∫[0,5] r dr dθ
= 10 * (1/2) * (25) * (2π)
= 250π.
Therefore, the value of the surface integral curl F · dS using Stokes' Theorem is 250π.