Verifying stoke's theorem

[mjf67089]

Homework Statement

Verify stoke's theorem for F(x,y,z)= y (i) where S is the hemisphere x^2+y^2+z^2=1, z is greater than or equal to zero, and the hemisphere is oriented in the direction of the positive z-axis.

Homework Equations

The equations that would apply here would be the equations for the two integrals to be computed, the Line integral: F(dr) and of the flux: curl(F)(ds).

The Attempt at a Solution

I think I have solved this problem but would like input on if my answer is right or wrong. If wrong, I'd appreciated the right answer and how to find it or hints. Here is my work:

parametrization: x=cos(t); y=sin(t); z=0
f(r(t))= (sin(t),0,0); dr= (cos(t))dt
F(r(t)) x dr = sin(t)cos(t)dt

the integral from 0 to 2(pi) sin(t)cos(t) equals (sin^2(t))/2. Plugging in 2pi and 0 one gets that this integral is equal to 0.

Now, for the second part of the verification process.
Curl = i j k
d/dx d/dy d/dz
y 0 0
This set up leads curl to be equal to: (0(x),0(y),-1(k)).
To find Rx x Ry, I remembered the parametrization that I set up, that is to say Rx x Ry is
(sin(t),0,0). Once you dot these two equations you get 0 so you know this second integral is equal to 0 as well.

Thus it appears that I have verified Stoke's theorem as both integrals are 0. Am I correct?

 

[mighty2000]

it is important to always double check your work and make sure your calculations are correct. In this case, your solution appears to be correct. However, there are a few things you can do to improve your work and make your solution more clear.

Firstly, it would be helpful to include a diagram or visualization of the hemisphere and the direction of the positive z-axis. This can help the reader better understand the problem and the orientation of the surface.

Secondly, when parametrizing the surface, it is important to check that the parametrization actually covers the entire surface. In this case, the parametrization x=cos(t), y=sin(t), z=0 only covers the top half of the hemisphere. To fully parametrize the surface, you would need to include the bottom half as well, which can be done by setting z=-sqrt(1-x^2-y^2). This ensures that the entire surface is covered and the parametrization is valid.

Additionally, it would be helpful to show the steps of your calculations for both integrals, rather than just stating the final answer. This can help the reader follow along and understand your thought process.

Overall, your solution appears to be correct. However, including a diagram, checking the parametrization, and showing your calculations in more detail can make your solution more clear and easy to follow.