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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.

Mathematics
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ya
Could you explain why please?
Something tells me that it is not right.

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Other answers:

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.
sorry here is the problem with proper notations
1 Attachment
I would agree with that yes, for every closed curve C. Path Independance.
if it is not a closed curve, would the second part still hold true?
No in this case it's path independent, but it's integral depends on the value of the end points.

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