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this post was submitted on 06 Jun 2025
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The issue here is that the operation of derivation is not an equivalence tranformation. Equivalence transformations must be reversible, but reversing the derivation would be an integration. But the integration introduces an integration constant of which the actual value is unknown. So, by deriving and integrating one expression, we basically add a constant to it and have changed it. This is why the derivation is no equivalence tranformation and messes up our equation.
this true, but in physics (and in maths too, especially in real analysis), we use differentiation as a equivalent transformation, in a generalised sense. we often come up with ways to define it with something like - finitely many discontinous points or something, which we can iron it. but major problem here (i think) is order of operation - we can not interchange sum and diff when they both depend on x