Stokes Theorem Problem: Surface Integral on Ellipse with Curl and Normal Vector

In summary, the conversation discusses the use of Stoke's theorem and Green's theorem to convert a surface integral into a line integral and then a double integral over an ellipse-shaped boundary curve. The final solution is pi.
  • #1

Homework Statement


F = xi + x3y2j + zk; C the boundary of the semi-ellispoid z = (4 - 4x2 - y2)1/2 in the plane z = 0

Homework Equations



(don't know how to write integrals on here, sorry)

double integral (curl F) . n ds

The Attempt at a Solution



curl F = 3y2x2k
n = k

curl F . n = 3y2x2

So I have a surface integral, which I think I can change to dA since the differential of the surface area is just 1dA...

Now this is where I'm stuck. How do i do the double integral with an ellipse? I tried it in rectangular coordinates but got some function I don't know how to integrate. Help!:confused:
 
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  • #2
Can you do this by integrating F dot dr over the boundary curve instead?

r(theta)=cos(theta) I+2sin(theta) J+0 K
 
  • #3
What function do you get that you can't integrate? It looks like it's just a trig substitution to me.
 
  • #4
what you can do is use Stoke's theorem to convert to the line integral of the vector filed and then if you use Green's theorem to convert that to a double integral, use a trig sub and then integrate. (integration takes a little bit of work) I think the answer is pi. Anybody agree?
 
  • #5
Yes, it's pi. Both ways.
 

1. What is Stokes Theorem problem?

Stokes Theorem is a mathematical theorem that relates the surface integral of a vector field over a surface to the line integral of the same vector field along the boundary of the surface. It is named after the mathematician George Gabriel Stokes.

2. What is the significance of Stokes Theorem problem in science?

Stokes Theorem is used in many areas of science, particularly in the fields of physics and engineering. It allows for the calculation of the total flow of a vector field through a surface by only considering the boundary of the surface. This makes it a powerful tool for solving problems involving fluid flow, electromagnetism, and other physical phenomena.

3. How is Stokes Theorem problem related to Green's Theorem?

Stokes Theorem is a higher-dimensional version of Green's Theorem. Green's Theorem relates a line integral around a simple closed curve to a double integral over the region enclosed by the curve. Stokes Theorem extends this idea to relate a surface integral over a surface to a line integral around the boundary of the surface.

4. What are the necessary conditions for applying Stokes Theorem?

There are two main conditions that must be met in order to apply Stokes Theorem: the surface must be a closed surface, meaning it has no boundaries, and the vector field must be continuously differentiable over the surface. Additionally, the surface must be oriented consistently with the direction of the line integral.

5. How is Stokes Theorem problem solved?

To solve a Stokes Theorem problem, first, determine if the necessary conditions are met. Then, evaluate the line integral along the boundary of the surface and the surface integral over the surface. If the two integrals are equal, this confirms that Stokes Theorem holds and the problem is solved.

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