Seemingly conflicting notions of a function

Throughout my mathematical education, I have seen a few, seemingly, different and conflicting notions of what a function is:

1. A function is a a type of mathematical object that maps every element of a set to some member of another set. E.g. $f:\mathbb{R} \rightarrow \mathbb{R},\ x \mapsto 2x$ is a function that maps every real number to another real number. Namely, $f$ maps every $t \in \mathbb{R}$ to $2t$ or $f(y) = 2y, \ y \in \mathbb{R}$.
2. If the value of a quantity, $a$, is written in terms of other quantities, $b$ and $c$; then $a$ is called a function of $b$ and $c$. E.g. If mass is constant, the force in $F=ma$ is called a function of acceleration and is written $F(a)=ma$. How does this relate to the notion of function in 1? The input $a$ refers to the value of a physical quantity here whereas in 1, the input is just a placeholder for an arbitrary real number so $f(a)=2a, \ a \in \mathbb{R}$ is the same as $f(m)=2m, \ m \in \mathbb{R}$.
3. There appears to be a dependence on the symbols used to represent the inputs when taking partials derivatives of functions. For example, $f(x)=2x$ and $f(y)=2y$ is the same mapping according to 1 but the partial with respect to $x$ of both of these are quite different.

How do you reconcile all of these ideas? What am I missing?

Thanks in advance

Solution 1:

Basically, #1 is what a function is, and you're confused about the context/meaning of things for #2 and #3.

For #2, if you wanted to give a formal meaning to the discussion, you could say something like the following (there are many variants that might do a better job with the physics):

In 1 spatial dimension, the triples of real numbers $(a,m,F)$ that represent physical situations (with the first coordinate representing acceleration, the second representing mass, and the third representing a force) are those that satisfy $F=ma$. For each value of the second coordinate $m$, there is at most one value of the third coordinate $F$ for a given value of the first coordinate $a$ (see the vertical line test). So for each value of $m$, we can define a function $F_m:\mathbb R\to\mathbb R$ so that: for all $a$, $\left(a,m,F_m(a)\right)$ is physical.

For #3, you're a little confused about the notation. $\dfrac{\partial}{\partial y}[\text{expression}]$ is equivalent to something like:

Use $[\text{expression}]$ to denote a function of at least two real variables, like $f:\mathbb R^7\to \mathbb R$, with the convention that $x$ is shorthand for $\operatorname{proj}_1$ (the first projection function outputting the first coordinate of the input), $y$ is shorthand for $\operatorname{proj}_2$, etc. Then take the partial derivative with respect to the second input, and rewrite everything with the $x,y,\ldots$ shorthand.

Since the function given by $f(y)=2y$ for all $y\in\mathbb R$ only has one input, we can't take its partial derivative with respect to the second input. And notation like $\dfrac{\partial}{\partial y}(2y)$ refers to a function like $f:(a,b)\mapsto 2b$ or $f:(a,b,c,d)\mapsto 2b$, etc.