Question

If a differential form is exact, when you change the variables does it remain exact?

Answer 1

Of course not.

Answer 2

I have the case of a central force \[\vec F = f(r)\hat r\]. The differential \[\vec F \cdot d\vec r\] is exact because you just have a potential in polar coordinates \[\int\limits^r f(r')dr'\], but does that mean that the form is exact also in cartesian coordinates? Because my book assumes that without proof

Answer 3

Yes. If the differential is exact, then its integral between any two points is independent
of the path(its a conservative force). that statement remains true regardless of what coordinates you use to describe the force.

Answer 4

OK, thanks