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Why are only functions discussed in physics and not relations?

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closed as unclear what you're asking by StephenG, ACuriousMind Jul 25 at 16:24

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Every function is also a relation. (But not the converse). Also, calculus is easy on functions because 1. The domain is exhausted, 2. There are no one-to-many mappings. Considering that most of modern classical physics started off with developments in calculus, this is not very surprising.

In modern physics people tend to implicitly use distributions and call them functions instead - the way some think of functions in physics slightly differs from that in mathematics, mostly because of often imposed strong conditions of continuity, differentiability, analyticity etc. In this light, it is a bit like topologists calling functions mappings.

Another point to consider is that physical models is often "just mathematics with extra steps" as Rick and Morty would have put it, so essentially adding interpretation to mathematical concepts and identifying them with physical entities and phenomena is one of the essential parts of the job. Relation, contrary to a function, implies an interpretation of association, not of prescription/construction. It's hard to think of electric field as a relation between space (a topological space or whatever you work with) an a vector space; instead, it seems more appropriate to assign elements of space values, which is using functions.

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  • $\begingroup$ Can calculus work on relations? $\endgroup$ – anubhav Jul 25 at 16:16
  • $\begingroup$ Calculus will work as long as you define it so it works. Some essential things to consider when approaching this construction would be sequences, limits, sums, appropriate measure (for Lebesgue) or Euclidean space structure (for Riemann). Then go on to define derivatives, integrals and hopefully prove some interesting theorems that utilize the features of relations that are not there with functions :) $\endgroup$ – xletmjm Jul 25 at 16:23

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