Stokes' Theorem for Line Integrals on Closed Curves: A Problem Solution

In summary, the conversation discusses using Stokes' theorem to find the line integral of F . dr over a closed curve C, which is the intersection of two surfaces. The integral can be simplified to \iint (curl F) \cdot \b{n} dA, where the normal vector n is \frac{1}{\sqrt{2}}(\b{i} + \b{j}) and the area is \pi r^2, with r being the radius of the circle formed by the intersection of the two surfaces. The final answer for the line integral is -2\sqrt{2}\pi.
  • #1
bigevil
79
0

Homework Statement



Please help me to check whether I did the right working for this problem. Thanks. The numerical answer is correct but I'm not very sure if the working is correct also.

Find [tex]\int y dx + z dy + x dz[/tex] over the closed curve C which is the intersection of the surfaces whose equations are [tex]x + y = 2[/tex] and [tex]x^2 + y^2 + z^2 = 2(x+y) [/tex]

The Attempt at a Solution



First, I note that the integral required is the line integral for F . dr where F = (y, z, x). Since the curve is closed, we can apply Stokes' theorem. By Stokes' theorem [tex]\int F . dr = \iint (curl F) \cdot \b{n} dA[/tex].

Curl F = (-1,-1,-1) after applying the cross product.

Then I sketch the surfaces on the x-y axis and pick out the normal vector [tex]n = \frac{1}{\sqrt{2}}(\b{i} + \b{j})[/tex]. Also [tex]\iint dA = \pi r^2[/tex]. Then the answer for the line integral is [tex]-2\sqrt{2}\pi[/tex]
 
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  • #2
What surface, exactly, are you integrating over? You are told that the curve is the intersection of two surfaces. To use Stoke's theorem, you must integrate over the entire surface: both of the given surfaces.
 
  • #3
It looks like bigevil is integrating over the plane. You don't have to integrate over both surfaces. Either one should give you the same answer. I'm a little confused at you deduced the radius of the circle. But other than that, it's fine.
 
  • #4
Sorry Halls, the closed curve is the intersection of the two equations. That curve is closed and the surface it covers is a circle (of radius sqrt(2) I think) then I'm integrating over the plane.

I don't know how to describe it, but I deduced by imagining that the line slices the sphere given. (I drew the whole thing in the x-y plane, ie looking from the 'top' down.) The plane (y + x = 2) runs through the centre of the sphere. I got radius sqrt 2 for the radius of the closed surface.

Thanx Dick and Halls.
 
  • #5
Yes, exactly. The plane cuts through the center of a sphere with radius sqrt(2). You can use Stokes over that surface. As you did so well.
 

FAQ: Stokes' Theorem for Line Integrals on Closed Curves: A Problem Solution

1. What is Stokes theorem and how is it used?

Stokes theorem is a fundamental theorem in vector calculus that relates the surface integral of a vector field over a closed surface to the line integral of the same vector field along the boundary of the surface. It is used to evaluate surface integrals in 3D space and has applications in fields such as electromagnetism, fluid mechanics, and differential geometry.

2. How do you apply Stokes theorem to solve a problem?

To apply Stokes theorem, you first need to identify the surface and the boundary curve in the problem. Then, calculate the curl of the vector field and evaluate the line integral along the boundary curve. Finally, plug in the values into the formula for Stokes theorem and simplify to get the final result.

3. What are the key assumptions in Stokes theorem?

The key assumptions in Stokes theorem are that the surface is closed, smooth, and orientable, and the vector field is continuously differentiable throughout the surface.

4. Can Stokes theorem be applied to non-closed surfaces?

No, Stokes theorem can only be applied to closed surfaces. For non-closed surfaces, the corresponding theorem is called the generalized Stokes theorem, which takes into account the non-closed nature of the surface.

5. What are some common applications of Stokes theorem in real-world problems?

Stokes theorem has many applications in physics and engineering, such as in calculating the circulation of a fluid around a closed loop or the flux of an electric field through a closed surface. It is also used in the study of fluid flow, electromagnetism, and differential geometry.

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