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and when you go from start to finish
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the total change will in fact be the change in U, from start to finish,
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and won't depend on how you got there.
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So basically, what you're doing is,
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in the xy plane
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you are defining a function U of xy,
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which is like a height.
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You defined the force in a clever way
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so that F·dr is the change in the height, as you march.
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So, you can start at any point on the mountain,
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go to any other point on the mountain;
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if you're only keeping track of the change in height