Understanding Independence of Path in Calculus 3

[livenn]

Hi,
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.

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.

 

[Tomsk]

OK I've done a crappy paint drawing, I hope it helps:

What it is saying is, for a conservative field, the integral over C1 = the integral over C2 = int over C3 = int over C4, because the value over the integral only depends on the end points R1 and R2, ie it is I(R2)-I(R1), where I'=F.

Think of a particle moving between two potential surfaces. If the force is conservative, and if the particle moves along an equipotential surface, no work is done on it. If it moves between potential surfaces, the work done on it only depends on the amount it has moved along the gradient of the potential. So you might as well only consider where it starts and where it ends, to find the distance along the potential gradient it has moved.