What could be causing issues with Green's Theorem?

[Chronocidal Guy]

Questions about Green's Theorem

When asked to demonstrate that Green's Theorem works, I keep coming up with disagreeing answers. I've been able to do the double integrals for the area of a region over a function F just fine, but whenever I try to integrate the same function over the closed curve using the normal vector, I keep getting zero... Say, I was integrating a function over a square. The double integral over the area works out just fine, but when I break the curve up into 4 sides, and try integrating each of those, I keep getting zero, because the opposite sides of the square cancel each other out. I don't really know what I'm missing, and it's been driving me nuts. Is there anything obvious that I'm just missing somehow?

 

[mathwonk]

yeah, the function has different values on opposite sides of the square so why should they cancel?

 

[M98Ranger]

There could be a few reasons why you are encountering problems with Green's Theorem. One possible issue could be that you are not using the correct orientation for the normal vector when integrating over the closed curve. The orientation of the normal vector should follow the right-hand rule, where your fingers curl in the direction of the curve and your thumb points in the direction of the normal vector. If you are not using the correct orientation, it could result in the opposite sides of the square canceling each other out.

Another potential issue could be that your function F is not continuous or differentiable over the entire region. Green's Theorem only applies to smooth functions, so if your function has any discontinuities or sharp corners, it may not work.

It's also possible that there could be a mistake in your calculations. Double check your work and make sure you are using the correct limits of integration and properly evaluating the integrals.

If you are still having trouble, it might be helpful to seek out a tutor or consult with your professor for further clarification on the concept. Sometimes having a fresh perspective can help identify any mistakes or misunderstandings. Don't get discouraged, Green's Theorem can be a tricky concept to grasp, but with practice and persistence, you will eventually get the hang of it.