2022-07-02

Is there a difference between "is proportional to" and "is a function of"?
If I say that $J$ (the volume current density) is proportional to $s$, the distance from the center axis of a long cylindrical wire, I can say that as $s$ changes, so does $J$
I feel as though I could say the same thing as $f\left(x\right)={x}^{2}$. As $x$ changes, $f$ also changes (according to the specified relationship, $x$ raised to the power of $2$).
Could someone help me clarify?

Alec Blake

Expert

The statement that some quantity $J$ is proportional to another quantity $s$ is equivalent to the statement that there exists some unspecified constant $k$ such that
$J=ks.$
This is the same as saying that $J\left(s\right)$ is the function $J\left(s\right)=ks$, so proportional to implies is a function of. However, a general function $J\left(s\right)$ need not satisfy a proportionality relation. For example,
$J\left(s\right)={e}^{s}$
is an example where $J$ is a function of $s$, yet $J$ is not proportional to $s$. Hence, being proportional to is a strictly stronger statement than being a function of.

malalawak44

Expert

Proportionality implies a linear relationship.
A function is much more general. The function $f\left(x\right)={x}^{2}$ is quadradic, not linear.

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