How do you know if a vector field is path independent

First notice that if you rotate the pictures by 180 degrees you'll get the same answer for line integrals by definition (the line integral is the same no matter how you rotate the answer).So let's call this c1, so this is c1, and this is c2.In other words, line integrals are path independent. given a conservative field and a closed curve, let a and b be any two points on the curve.The line integral of a conservative field depends only on the value of the potential function at the endpoints of the domain curve.Conservative fields are independent of path.

I'll give a different answer to alon's because it illustrates a different set of ideas.Not sure if irrotational implies conservative, but conservative implies irrotational, so that if it's not irrotational, it's not conservative.(technically the argument above assumed that \(c\) was smooth, but we can replace \(c\) by a piecewise smooth curve by splitting the line integral up into the sum of finitely many.If this is the case, then the line integral of f along the curve c from a to b is given by the formula.