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It's possible to prove with mathematical terms using a part of the physics knowledge this general assumption? I have several formulas in mind but not a general and big picture for this property of the integrals, and the most important part is that i can't find a rigorous explanation for that.

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What integral? What question? – J. M. Dec 1 '11 at 12:52
what are your doubts? I'm considering a generic integral, an infinite sum of infinitesimals. – Micro Dec 1 '11 at 12:55

Sorry, but your question does not make sense. First we'll have to fix the term "integral" we are talking about.

Then we'd have to fix the physical theory we talk about, like classical mechanics. A potential function in classical mechanics is a function from physical space and time to the reals. The (negative of the) gradient of this function is the force, the line integral (Riemann integral) of this function (along a path between two fixed points) is proportional to the energy (difference of a point mass located at the two fixed points) with respect to this force.

In a static situation where the force does not change with time, but does depend on space coordinates only, the energy difference of two points must be independent of the path we choose to integrate along. This is a condition (on the potential function) that is necessary in order for the physical interpretation to make sense. If a function does not satisfy this condition, we cannot accept it as a representation of a potential in our physical theory (classical mechanics in this case).

Am I close to anything you were thinking of?

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You are close to my topic but not to what my teacher says. How many definitions are available for the integral operator ( i'm referring this question to the first line of your reply )? We never have treated the potential as a function but as system, a linear system that sometimes we develop with a matrix or with classic algebra calculus, and the integral gives us a quantity that can rapresent the total amount of energy available in a system, i am not talking about law or function, i only consider the integral as an operator that can quantify the potential as result. Where i'm wrong? – Micro Dec 2 '11 at 2:47

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