Line Integral of a Vector Field over a Half Sphere using Stoke's Theorem

[sunnyday11]

Homework Statement

F = ( 2y i + 3x J + z² k where S is the upper half of the sphere x² + y² + z² = 9 and C is its boundary.

Homework Equations

The Attempt at a Solution

I used Stoke's Theorem and found the solution to be 36 pi, but when I use line integral to verify, using substition:

r(t) = 3cost i + 3sin t j

F(r(t))r'(t) = 9 (-2sin²t + 3cos²t)

\int^{2 pi}_{0} 9 (-2sin²t + 3cos²t)

= 9 (1/2 + 5/2 cos(2t))/^{2 pi}_{0}

= 9 pi which doesn't agree with 36 pi.

I wonder if I did my substitution wrong. Should I do anything with z or the k component?

Thank you!

 

[Sethric]

The math looked right to me, so I did the curl side and parametrized the surface. I found the answer to be 9 \pi both ways. Check your curl, and make sure you are actually dealing with the curl vector dotted with the normal vector to the surface.

 

[sunnyday11]

Thank you!

Yes I forgot to consider the normal after computing the curl. So I get to surface integral of 1 which translates into double integral of 1 over the region enclosed by S and since x and y form a circle the region is the area of a circle which is 9 pi.