# Equations and functions in physics and mathematics

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?

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 $$x$$.