Function notation in physics equations

Main Question or Discussion Point

I have a question about function notation as it is used in physics equations.

Say that we have the equation

$$x = x_{0} + v_{0}t + \frac{1}{2}at^{2}$$

Obviously, x is a variable quantity representing all possible displacement values that are dependent on the value t. But, what happens when we do

$$x(t) = x_{0} + v_{0}t + \frac{1}{2}at^{2}$$

Does x then become a function rather than a variable? Typically, in function notation, we denote f as the function and f(x) as the variable output that depends on x.

So in the case of the above equation, how do we reconcile that x constantly changes from a variable quantity to the name of a function?

jedishrfu
Mentor
In both cases, x is a dependent variable. In the first case, it's implied to be t and in the second its explicitly stated.

Mark44
Mentor
In both cases, x is a dependent variable. In the first case, it's implied to be t and in the second its explicitly stated.
To clarify a bit what jedi said, in the first equation, it is implied that t is the independent variable, while this explicit in the second equation.

To clarify a bit what jedi said, in the first equation, it is implied that t is the independent variable, while this explicit in the second equation.
Doesn't this contradict what function notation usually says though, that in f(x) f is the symbolic representation of a function, which means it's not a dependent variable. Do we say that " in x(t), x is the dependent variable" just by convention?

Bystander
Homework Helper
Gold Member
Just googled SIAM (Society of Industrial and Applied Mathematics) and mathematical notation on the suspicion that this has been covered somewhere a la IUPAC an IUPAP, and got
http://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style/Mathematics
which might help resolve some of the questions/issues.
I didn't find anything in the article regarding function notation.

Bystander
Homework Helper
Gold Member
Copy this, paste that. Too many bells & whistles in my browser. http://en.wikipedia.org/wiki/Function_(mathematics) Was what I meant to paste, but going over it just now, it does NOT draw on SIAM. Speed read both, conflated SIAM reference from what just disappointed you with content of this second. There may not be any more convention on this than there is on IUPAC and naming organic compounds.

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I have a question about function notation as it is used in physics equations.

Say that we have the equation

$$x = x_{0} + v_{0}t + \frac{1}{2}at^{2}$$

Obviously, x is a variable quantity representing all possible displacement values that are dependent on the value t. But, what happens when we do

$$x(t) = x_{0} + v_{0}t + \frac{1}{2}at^{2}$$

Does x then become a function rather than a variable? Typically, in function notation, we denote f as the function and f(x) as the variable output that depends on x.

So in the case of the above equation, how do we reconcile that x constantly changes from a variable quantity to the name of a function?
You are correct. An extremely pedantic mathematician would write x = f(t) and then define f to be the function defined by $f(t) = x_{0} + v_{0}t + \frac{1}{2}at^{2}$, for all $t\geq 0$, where a function is a rule or set. However, such abstraction is really unnecessary at this level of physics, so it is common to conflate variables with function names. Thus we get notation like x = x(t), where the x on the left side is a variable, an element of the image of a function, while x on the right side is a function, a set or rule.
Physicists tend to be rather flippant in notation, so you can expect more conflation in the future. Mathematicians tend to inspect the structure of operations in more detail than would necessarily arise in a physical application, which requires the more precise language that separates the value of a function from a function. As such, expect vectors to be conflated with vector fields, and likewise for tensors and tensor fields.

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Only mathematicians really care about the correctness of notation. In the case you presented,

$x(t) = \frac{1}{2}at^2 + v_0 t + x_0$

$x_0$ is just the initial value for the function, just like you might write a linear equation like:

$y(x) = ax + y_0$

Though I don't know how many people actually do that; but $x$ is the function value at time $t$, so $x_0$ is like saying, "$x(0)$", if you like. You could replace it by anything sensible, like "$s_0$", or "$p_0$". Just as long as you don't think that $x$ is the independent variable.

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SteamKing
Staff Emeritus
Homework Helper
There's no hard and fast rule, in my experience, which states that functions must only be designated using an 'f', e.g., f(x), f(t), etc.

Working with polar coordinates and vectors, often the position function is designated r(θ) or r(t), for example.

In working with electricity, the current can be a function of time, and its function is designated as i(t); similarly, the voltage function can be designated v(t).

In working with the structural mechanics of beams, the shear force, bending moment, slope, and deflection functions w.r.t. position are indicated V(x), M(x), Θ(x), and δ(x), respectively, quite frequently.

As long as the notation is explicitly defined and clear to the reader, you can use whatever letter to indicate a function, as long as it is distinct from any independent variables on which it depends.

Stephen Tashi
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