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To tie some intuition back, here's how I(region, derivative(quantity))
= I(boundary(region), quantity)
is just the Fundamental Theorem of Calculus.What you need to do is take "region" to be an interval of the real line like [a, b]. Now, the boundary of "region" is the set of the end points, {a, b}. Thus, we've now transformed this equation to I([a,b], d(quantity)) = I({a, b}, quantity)
If we see I as being a sum, it's clear that the right-hand-side must be the same as a regular sum, though we need to account for the idea that we're summing from a and to b. Ultimately, we do this by negating where we're coming from[1] I([a,b], d(q)) = -q(a) + q(b)
Then we just have to recognize I([a,b], _) as the definite integral from a to b DefiniteIntegral(a, b, d(q)) = q(b) - q(a)
and this is a statement that the definite integral of a function on an interval [a,b] is the difference of the antiderivative of that function evaluated at the endpoints---the standard fundamental theorem of calculus!But notice that in this transformation we've destroyed some information. No longer is it so apparent that "boundary" and "derivative" have some kind of kinship. We also cannot easily generalize this notion to higher dimensional spaces (unless we already know the Stokes Law trick). [1] This is a bit arbitrary. If we chose it the other way the equation would still hold, it'd just not be the standard convention and thus less recognizable. The reason we choose this is because calculus, it turns out, depends upon a notion of orientation. We have to know whether we're going with or against "the flow". |