The discussion revolves around the application of Green's Theorem to evaluate a line integral along a positively oriented curve, specifically an ellipse defined by the equation x² + xy + y² = 4. The integral in question is ∫_{C} sin(y)dx + xcos(y)dy.
The conversation includes affirmations of the original poster's reasoning, along with corrections regarding the conditions for the line integral to be zero. Some participants emphasize the importance of understanding that not all line integrals around closed curves yield zero unless specific conditions are met.
There is a focus on the nature of the vector field involved, with participants noting that the field in question is conservative, which influences the outcome of the integral. The discussion highlights the potential for misunderstanding regarding when line integrals are zero.
gomunkul51 said:Correction: The line integral around any closed curve will be 0 ONLY if the integral is on a conservative field. otherwise it won't be zero for all cases.
Dick said:Mmm. Well, sure. F=(sin(y),x*cos(y)) is conservative. So the integral of F.dr is zero around any closed curve. I didn't mean ANY F. Did that really need a 'correction'?