Is there a difference between "is proportional to" and "is a function of"?If I say...





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 ( x ) = 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?

Answer & Explanation

Alec Blake

Alec Blake


2022-07-03Added 11 answers

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 = k s .
This is the same as saying that J ( s ) is the function J ( s ) = k s, so proportional to implies is a function of. However, a general function J ( s ) need not satisfy a proportionality relation. For example,
J ( s ) = 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.



2022-07-04Added 4 answers

Proportionality implies a linear relationship.
A function is much more general. The function f ( x ) = x 2 is quadradic, not linear.

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