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Introduced in Prof. S--'s precalculus class on Monday was an analogy to a car muffler in describing a function ("something going in, something coming out"). This approach helps draw a clearer distinction between the "set of ordered pairs"
f = { (x,y) | ... conditions that guarantee non-repeating first coordinates ... }
and the inputs or outputs viewed in isolation, in the same way that edges in a graph are distinguished from vertices. When we let the idea of a "set of ordered pairs" fade into the background, as Leibniz notation encourages us to do, and use the same letter for the second coordinate as for the rule of association, then we plant the seeds for later confusion when calculating partial derivatives or executing a change-of-coordinates.
An example from an early lecture in a quantum mechanics course: Suppose a particle moves in the xy-plane subject to the potential V(x,y) = x^2+y^2. What is V(r,theta)?
Half the class comprised physics majors, for whom the standardized letters for physical quantities led them to suspect that the question meant for them to express the potential as a function of polar coordinates, and they answered accordingly V(r,theta) = r^2.
Other students with more exposure to computer science resisted the temptation to overload the letter V with the conflicting interpretations of function (rule of association) and physical potential (output of the function), so they answered V(r,theta) = r^2+theta^2.
Without a consensus on how to answer this precalculus question, it's easy to imagine the profusion of inconsistent responses when the class is asked to find \frac{\partial V}{\partial r} or \frac{\partial V}{\partial x}.
Even in the context of today's circular ripples problem, where we wrote both A(r) = pi*r^2 and A(t) = 9*pi*t^2, the students might justifiably arrive at two different answers when asked to find, say, A(2) or A'(5). In practice we usually manage to avoid this problem by providing units for the independent variable, so that students can select the correct expression for A or A' using dimensional analysis.