Question:
Compute the line integral for the vector field $F(x,y) = (x^2y,y^2x)$ and the path $$r(t) = (\cos t,\sin t),\quad t \in [0,2\pi]$$
Answer:
The answer I am getting is $0$, I am fairly certain this is the correct answer. What is the meaning behind a $0$ value for a line integral. Does that mean the line doesn't exist? Or the path is symmetric?
Best Answer
It doesn't mean anything special. Maybe one could have detected that the value of the integral is $0$ using symmetry considerations, making the actual computation of the integral superfluous.
Things start to get interesting if the integral is $0$ for all closed curves, not just a particular one. A necessary condition for this would be that your $F=(P,Q)$ satisfies $${\partial Q\over\partial x}-{\partial P\over\partial y}\equiv0\ .$$ Since this is not the case not much more can be said in connection with this example.