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 JI 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?

Question

Is there a difference between &quot;is proportional to&quot; and &quot;is a function of&quot;?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   JI 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?

Alec Blake · Accepted Answer

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.

malalawak44 · Accepted Answer

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