Use Stokes' Theorem to evaluate
S
curl F · dS.
F(x, y, z) = 2y cos z i + ex sin z j + xey k,
S is the hemisphere x2 + y2 + z2 = 49, z ≥ 0, oriented upward.
To use Stokes' Theorem to evaluate the integral ∫∫_S curl F · dS, we need to follow these steps:
Step 1: Compute the curl of F.
Step 2: Find the unit normal vector to the surface S.
Step 3: Determine the appropriate parameterization of the surface S.
Step 4: Calculate the dot product between curl F and the unit normal vector.
Step 5: Evaluate the surface integral using the dot product calculated in step 4.
Now let's perform these steps one by one.
Step 1: Compute the curl of F.
The curl of F is given by:
curl F = (∂Fz/∂y - ∂Fy/∂z) i + (∂Fx/∂z - ∂Fz/∂x) j + (∂Fy/∂x - ∂Fx/∂y) k.
Let's calculate the partial derivatives first:
∂Fx/∂y = 0,
∂Fx/∂z = -2y sin(z),
∂Fy/∂x = ey,
∂Fy/∂z = 0,
∂Fz/∂x = ey,
∂Fz/∂y = 2cos(z).
Therefore, the curl of F is:
curl F = (2cos(z) - ey sin(z)) i + ex sin(z) j + (ey - 0) k.
Step 2: Find the unit normal vector to the surface S.
The unit normal vector to the surface S can be obtained by taking the gradient of the equation of the surface:
n = grad(x^2 + y^2 + z^2) / |grad(x^2 + y^2 + z^2)|,
where grad denotes the gradient and | | represents the magnitude.
Taking the gradient, we get:
grad(x^2 + y^2 + z^2) = 2x i + 2y j + 2z k.
The magnitude of the gradient is:
|grad(x^2 + y^2 + z^2)| = sqrt(4x^2 + 4y^2 + 4z^2) = 2sqrt(x^2 + y^2 + z^2).
Dividing the gradient by its magnitude, we obtain:
n = (x i + y j + z k) / sqrt(x^2 + y^2 + z^2).
Since we are given that the surface S is the hemisphere with x^2 + y^2 + z^2 = 49 and z ≥ 0, the unit normal vector is (x/7) i + (y/7) j + (z/7) k.
Step 3: Determine the appropriate parameterization of the surface S.
To parameterize the surface S, we can use the spherical coordinates:
x = 7 sinφ cosθ,
y = 7 sinφ sinθ,
z = 7 cosφ,
where 0 ≤ φ ≤ π/2 and 0 ≤ θ ≤ 2π.
Step 4: Calculate the dot product between curl F and the unit normal vector.
Substituting the parameterization of the surface S into curl F and the unit normal vector, the dot product becomes:
(curl F · n) = [(2cos(z) - ey sin(z)) (x/7) + ex sin(z) (y/7) + (ey) (z/7)].
Replacing x, y, and z with their respective expressions in terms of φ and θ:
(curl F · n) = [(2cos(7cos(φ)) - e(7sin(φ)) sin(7cos(φ))) (7sin(φ) cos(θ)/7) + (e(7sin(φ)) sin(7cos(φ))) (7sin(φ) sin(θ)/7) + (e(7sin(φ))) (7cos(φ)/7)].
Simplifying:
(curl F · n) = (2cos(7cos(φ)) - e(7sin(φ)) sin(7cos(φ))) sin(φ) cos(θ) + (e(7sin(φ)) sin(7cos(φ))) sin(φ) sin(θ) + (e(7sin(φ))) cos(φ).
Step 5: Evaluate the surface integral using the dot product calculated in step 4.
Now we integrate this dot product over the surface S:
∫∫_S curl F · dS = ∫∫_S [(2cos(7cos(φ)) - e(7sin(φ)) sin(7cos(φ))) sin(φ) cos(θ) + (e(7sin(φ)) sin(7cos(φ))) sin(φ) sin(θ) + (e(7sin(φ))) cos(φ)] dS.
To evaluate this surface integral, we would need to provide specific limits for φ and θ.
To use Stokes' Theorem to evaluate the integral ∬ curl F · dS, we need to follow these steps:
Step 1: Determine the orientation of the surface S by checking the direction of the normal vector.
In this case, S is defined as the hemisphere x^2 + y^2 + z^2 = 49 with z ≥ 0, oriented upward. For this surface, the normal vector points outward from the surface and is in the positive z-direction.
Step 2: Compute the curl of the vector field F.
The curl of F is given by:
curl F = (∂Q/∂y - ∂P/∂z) i + (∂R/∂z - ∂P/∂x) j + (∂P/∂y - ∂Q/∂x) k
Given that F(x, y, z) = 2y cos z i + ex sin z j + xey k, we can identify P, Q, and R as:
P = 2y cos z
Q = ex sin z
R = xey
We can now calculate the partial derivatives:
∂P/∂x = 0
∂P/∂y = 2 cos z
∂P/∂z = -2y sin z
∂Q/∂x = ex sin z
∂Q/∂y = 0
∂Q/∂z = ex cos z
∂R/∂x = ey
∂R/∂y = x
∂R/∂z = 0
Substituting these values into the formula for the curl, we have:
curl F = (ex cos z - 2 cos z)i + (ey - ex sin z)j + (2y sin z - x)k
Step 3: Calculate the surface area element dS.
Since S represents a hemisphere, we can use spherical coordinates to parameterize the surface.
We have:
x = r sinθ cosφ
y = r sinθ sinφ
z = r cosθ
We can find the surface area element dS in terms of spherical coordinates by taking the cross product of the partial derivatives:
dS = |(∂r/∂θ x ∂r/∂φ)|dθdφ
The limits of integration for θ are from 0 to π/2 (since z ≥ 0), and for φ, the limits are from 0 to 2π.
Step 4: Set up the integral using Stokes' Theorem.
Using Stokes' Theorem, the integral ∬ curl F · dS can be written as:
∬ curl F · dS = ∭ (curl F) · dV
In this case, since S is a hemisphere, we can integrate over the volume enclosed by the hemisphere.
Using spherical coordinates, the volume element dV is given by:
dV = r^2 sinθ dr dθ dφ
Therefore, the integral becomes:
∬ curl F · dS = ∭ (curl F) · dV
= ∭ [(ex cos z - 2 cos z)i + (ey - ex sin z)j + (2y sin z - x)k] ·
[r^2 sinθ dr dθ dφ]
Step 5: Evaluate the integral.
Now, we proceed to evaluate the integral by performing the triple integration:
∬ curl F · dS = ∭ [(ex cos z - 2 cos z)i + (ey - ex sin z)j + (2y sin z - x)k] ·
[r^2 sinθ dr dθ dφ]
To simplify the calculation further, we need to convert the curl of F from Cartesian coordinates to spherical coordinates and express all terms in terms of r, θ, and φ.
Finally, evaluate the triple integral over the given limits of integration to find the result.
The curve for the line integral is just
x = 9cos(t)
y = 9sin(t)
z = 0
Now just evaluate F.dr and integrate for t in [0,2pi]