Stoke's Theorem Sphere and Plane

[IniquiTrance]

Homework Statement

Find the flux of the curl of F: $$\vec{F}=<yz,xz,xy>$$

Over S defined by:

Sphere: $$x^{2}+y^{2}+z^{2}=1$$

Where $$x+y+z \geq 1$$

Homework Equations

The Attempt at a Solution

I know I have to use Stoke's theorem to evaluate the line integral counterclockwise around the circular path formed by the intersection of:

$$x^{2}+y^{2}+z^{2}=1$$

and

$$x+y+z = 1$$

I'm having trouble setting up this line integral. Any insights would be greatly appreciated.

 

[mighty2000]

Thank you for your forum post. It seems like you are on the right track by using Stoke's theorem to evaluate the flux of the curl of the vector field over the given surface. To set up the line integral, you can first parameterize the circular path formed by the intersection of the sphere and plane. This can be done by setting x = cos(t), y = sin(t), and z = 1-cos(t)-sin(t), where t ranges from 0 to 2π.

Next, you can calculate the curl of the vector field F, which is given as <yz, xz, xy>. This can be done by taking the determinant of the Jacobian matrix of the vector field, which will give you <0, 0, 0>. This means that the flux of the curl of F over the given surface will be zero.

I hope this helps with setting up the line integral. Please let me know if you have any further questions. Good luck with your calculations!



Scientist