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livenn

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I'm having a bit of difficulty wrapping my mind around the concept of independence of path. My textbook says:

If F is continuous and conservative in an open region R, the value of int(F.dr) over the curve C is the same for every piecewise smooth curve C from one fixed point in R to another fixed point in R. This result is described by saying that the line intint(F.dr) over the curve C is independent of path in the region R.

If F is continuous and conservative in an open region R, the value of int(F.dr) over the curve C is the same for every piecewise smooth curve C from one fixed point in R to another fixed point in R. This result is described by saying that the line intint(F.dr) over the curve C is independent of path in the region R.

I get the continuous and conservative on the open region part...

But I'm failing to comprehend how this means that

*the value of int(F.dr) over the curve C is the same for every piecewise smooth curve C from one fixed point in R to another fixed point in R*

This really makes no sense to me, and I don't have a visual image of this to consult, could anyone enlighten me, or does anyone know of any good images or applets that explain this clearly? Unfortunately I haven't found any on google. Thanks in advance.

Edit: Just as a point of reference, this is for calculus 3.