My question has to do with the semantics of "proportionality". Often people say quantity y is proportional to quantity x if increasing x by a factor k increases y by the same factor. However, doesn't this simple "definition" lead to inconsistencies when it comes to differentiation/integration?
For example, the total charge Q in some volume is equal to the triple integral of the charge density over the volume. Doubling the charge density function doubles its integral over the volume and hence results in a total charge 2Q. Would it be appropriate to say Q is proportional to the charge density in that case? Other examples come to mind, such as the relationship between acceleration and velocity, and invoking the constant factor property of the derivative.
Thank you in advance.