In: Advanced Math
Let⇀F(x,y) =xi+yj/(e^(x^2+y^2))−1. Let C be a positively oriented simple closed path that encloses the origin.
(a) Show that∫F·Tds= 0.
(b) Is it true that ∫F·Tds= 0 for any positively oriented simple closed path that does not pass through or enclose the origin? Justify your response completely.
Dear student,
The problem is completely solved below.
Feel free to ask doubts in the comments section..
Justification:
For stokes theorem only you need to think is where the curve is CLOSED or not. Then only you will be able to form a surface integral which is not defined without a closed boundary. Here This theorem is not based on any reference point like origin or some other point.Also you MUST ensure the that is integrable (piece wise integrability is also accepted.)These are the only conitinous that stokes theorem demands.
The above curve satisfied integrability condition and closed.So it has freedom to get the theorem applied on it.
According to definition,
Here curl is zero irrespective of any condition.So you can always choose a closed path that may not pass through origin or may not contain origin in it.
If you feel still confused.Feel free to comment your doubt.