Question

Is it true that if F is a plane vector field such that ∮_c(F.dr)=0 for every closed curve C, then ∫_c(F.dr)=0 for every curve C? Please explain.

Answer 1

ya

Answer 2

Could you explain why please?

Answer 3

Something tells me that it is not right.

Answer 4

can you please check your opening post again @qwerty54321, is it all right written out? If F is a potential function then the gradient of that function with dr in a closed path will result to 0, that is true. But I have troubles understanding the post above.

Answer 5

sorry here is the problem with proper notations

Answer 6

I would agree with that yes, for every closed curve C. Path Independance.

Answer 7

if it is not a closed curve, would the second part still hold true?

Answer 8

No in this case it's path independent, but it's integral depends on the value of the end points.