Proof of F is conservative if delta_W = 0?

From: sweet (sweet430_at_hotmail.com)
Date: 06/17/04 

Date: 17 Jun 2004 07:37:57 -0700

I am reading Goldstein's mechanics and in the very first chapter he
states that if the work from pt. 1 to pt. 2 is the same for different
paths, then the force is conserved. I know that since delta_P = 0,
then mg is a conservative force. But I was wondering how to do show
that in general using the definition of W(1,2), namely:

Let W(1,2) := the work performed by some force F from point 1 to point
2.
Let W(1,2) := I(1,2) [ F.ds ].

Let I(1,2) := the integral from point 1 to point 2.
Let delta_I(1,2) := I(1,2)_path2 - I(1,2)_path1

And so the statement reduces to showing that delta_I(1,2) = 0. How
then does one prove that the force, F, is conservative? The furthest
one can go is:

I(1,2)[ (F.ds)_path2 - (F.ds)_path1 ] = 0,

since both F and ds are arbitrary. Then again, since F.ds = 0 for
perpendicular components... nah... that's only good for 1 specific
angle.

So perhaps you can see where I am stuck and help me out here.

Thanks in advance for any help you may give me.

-sweet
http://www.angelfire.com/ny5/jbc33/

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