Elliptic Line Integral: Solving for Circulation Around an Ellipse

In summary, the conversation discusses the evaluation of the line integral xdy - ydx over the ellipse with center (0,0), major axis of length 2a, and minor axis of length 2b. Two methods were used to solve this, one involving parameterization and the other using Green's theorem. Both methods resulted in the answer 2πab. However, there is confusion regarding the correct answer, with one book stating it as πab/2. The conversation concludes with the realization that the line integral should actually be -ydx + xdy, and therefore the correct answer is 2A, where A is the area of the ellipse.
  • #1
terhorst
11
0

Homework Statement


Let C be the ellipse with center (0,0), major axis of length 2a, and minor axis of length 2b. Evaluate [tex]\oint_C xdy - ydx[/tex].

Homework Equations


I solved this two ways. First I parameterized x and y as [tex]x=a \cos \theta[/tex] and similarly for y. I also applied Green's theorem, which yielded [tex]\oint_C xdy - ydx = 2 \int \int_D dA[/tex] where D is the area enclosed by C (ie an ellipse.) In both cases I got the answer [tex]2\pi a b[/tex].

The Attempt at a Solution


My only question is, the book I am using says the answer is [tex]\frac{\pi a b}{2}[/tex]. This is an ETS book and they don't usually have typos, especially when it's the answer key to a previously administered exam. What am I missing?
 
Physics news on Phys.org
  • #2
The answer is pi*a*b/2. If a=b=r then it's a circle and the area is pi*r^2. So the contour is half that.
 
  • #3
I apologize for being so dense, but I'm still confused. A couple different books I have print the result

[tex]\frac{1}{2}\oint_C -ydx + xdy = \iint_{R} dA = A[/tex]

If the area of the ellipse is [tex]A=\pi a b[/tex] then I would think that the value of the line integral is [tex]2A[/tex].
 
  • #4
Sorry, yes, I think you are right. Don't know what I was thinking...
 

1. What is an elliptic line integral?

An elliptic line integral is a type of line integral that is used in mathematics and physics to calculate the work done by a force field along a curved path. It takes into account the shape of the path and the direction of the force field, making it useful for analyzing complex systems.

2. How is an elliptic line integral different from a regular line integral?

An elliptic line integral takes into account the shape of the path, while a regular line integral only considers the endpoints of the path. This makes an elliptic line integral more accurate for curved paths, while a regular line integral is better suited for straight paths.

3. What are some applications of elliptic line integrals?

Elliptic line integrals are used in a variety of fields, including physics, engineering, and mathematics. They are commonly used to calculate the work done by electric and magnetic fields, as well as for analyzing fluid flow and modeling complex systems.

4. How is an elliptic line integral calculated?

To calculate an elliptic line integral, the path must be broken down into small segments, and the work done by the force field along each segment must be calculated. These individual work values are then added together to get the total work done along the entire path.

5. Are there any limitations to using an elliptic line integral?

One limitation of elliptic line integrals is that they can only be used for conservative force fields, where the work done is independent of the path taken. They also require the path to be smooth and continuous, and may not be suitable for highly complex or chaotic systems.

Similar threads

  • Calculus and Beyond Homework Help
Replies
12
Views
935
  • Calculus and Beyond Homework Help
Replies
3
Views
116
  • Calculus and Beyond Homework Help
Replies
12
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
826
  • Calculus and Beyond Homework Help
Replies
1
Views
449
  • Calculus and Beyond Homework Help
Replies
4
Views
954
  • Calculus and Beyond Homework Help
Replies
5
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
3K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
146
Back
Top