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In physics we can define velocity as the derivative of position. We can write:

$$u = \frac{d}{dt}x(t)$$

or

$$u = g(t)$$

where $g$ denotes the function after differentation of the position with respect to time. In mathematics there is a sublte difference between $g$ and $g(t)$. We should use $g$ to denote the object (the function) and $g(t)$ to denote the value of the function at a particular value of $t$. So should we say that velocity is $g$ or $g(t)$?

Another thing that is generally not clear is when we have a variable $y$ that depends linearly on another variable $x$. Which of the following ways is the correct to express this dependence and why?

$$ y = ax + b$$

or

$$ y(x) = ax + b$$

For example in ideal gas law:

$$pV=nRT$$

Should we treat $p$ as function of $(n, V, T)$?

I am using this example because in Thermodynamics "Implicit differentation" is used a lot.

Edit What also makes things more complicated is the fact that we can differentiate the left and right hand side of an equation. For example we could write:

$$\frac{dy}{dx} = \frac{d}{dx}(ax + b)$$

Differentiation only makes sense for functions. Is there a general rule to help distinguishing how we should express a dependence between two (or more) physical quantities? Are both ways equivalent?

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It is true that physicists are often very lax about distinguishing a function $f(.)$ and its value at a particular point $f(t)$. This is not the only place: e.g., physicists would often use the same symbol to denote a random variable $X$ and its values $x$.

In case of ideal gas, the state equation is usually understood in the context of a specific thermodynamic ensemble, which is characterized by a set of independent variables.

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    $\begingroup$ In addition, the physical constraints of a system that you are modeling add context that you don't get when doing "pure" mathematics. $\endgroup$ Sep 13 at 16:22
  • $\begingroup$ @DavidWhite agreed. $\endgroup$ Sep 13 at 16:23
  • $\begingroup$ @RogerVadim What about the relation between two physical quantities? Should we represent it as a function or as an equation? $\endgroup$
    – Anton
    Sep 16 at 17:55

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