2

I thought I understood what a proportionality was, but it was recently called into question. I want to make sure I fully understand how the term is generally used in the engineering world. The basic explanation of proportionality is that two variables are related by a constant multiplier such that a change in one of them results in a change in the other, scaled by this constant of proportionality. So we could say:

$$ A(x_1, x_2, ... x_n) \propto B(x_1, x_2, ... x_n) $$

or

$$ A(x_1, x_2, ... x_n) = cB(x_1, x_2, ... x_n) $$

where $A$ and $B$ are the proportional variables and $c$ is the constant of proportionality.

My question deals specifically with what restrictions are placed on this constant. In my previous description, $A$ and $B$ were functions of a number of variables, they are necessarily the same variables. Is this the case?

Additionally, in engineering, few things are every truly a constant, so our constant of proportionality could be a function of a number of parameters:

$$ A(x_1, x_2, ... x_n) = c(y_1, y_2, ... y_n)B(x_1, x_2, ... x_n) $$

However, it is my understanding that when discussing proportionality it must be assumed that all of those $y_n$ parameters must be held constant within your system. This is where my understanding starts to come into question. It was suggested that $c$ need not be independent of $A$ or $B$ and this is still a proportionality:

$$ A = c(B)B $$

Now, I had always assumed that this is a prime example of a situation where $A$ is not proportional to $B$. But upon thinking about it further, I came up with 3 limited domain cases given the following relationship:

$$ A = (e^B - 1)B $$

case 1: $B < -10$

case 2: $B > 10$

case 3: $-\infty < B < \infty$

It seems to me that for case 1, we could perhaps say that $A \propto B$ given the small change in the coefficient. But for cases 2 and 3, would it ever be reasonable to consider $A \propto B$?

To clarify my questions: Is anything about my understanding as outlined above wrong? Are there any situations where it is correct to say $A \propto B$ and that the constant of proportionality is a strong function of $B$? Does this answer change if we can consider $B$, and therefore $A$ to be nearly a constant? Is that a useful situation to consider?

Matt
  • 123
  • 3
  • if c depends on b (or on any x) in any way then you do not have proportionality. In the case where the dependence is weak you might use some terminology such as "nearly proportional" "approximately proportional" etc. – agentp Nov 12 '17 at 15:18

1 Answers1

3

Proportionality is a little more complex than that. In general, it only means they are correlated, and the correlation is monotonic.

Generally two variables are proportional, if $$A = f(B)$$

where f is a monotonic function.

  • The basic variant is "directly proportional" - a linear relation:

$$ A = cB + d $$

with constant (or independently parametric) $c$ and $d$; $c > 0$. Usually, when you talk about proportionality without other qualifiers, it's what is meant. $d$ is called offset or bias, and you may talk about biased proportionality when it's non-zero.

  • Then there's the very common "inversely proportional."

$$ A = {c \over B} + d $$

  • $A = cB + d$ for negative $c$ is "linearly proportional, with negative coefficient" - sometimes mislabeled as "inversely proportional". General $A = cB + d$ for non-zero $c$ with sign not given is just "linearly proportional".

Then there are others:

  • "quadratically proportional", $A = cB^2 (+dB +e)$

  • "exponentially proportional", $A = e^{cB} (+ d)$

  • "logarithmically proportional", $A = c \ log B (+ d)$

  • In other cases, you'll see "proportional to f of B", e.g. the name of $A = cB^4$ is not called "tesseractically proportional", just "proportional to fourth power of B".

But note all these functions are strongly monotonic. There's no such thing as "sinusoidally proportional".

SF.
  • 6,135
  • 24
  • 43
  • one more convoluted example: Force exerted by a permanent magnet on a piece of iron is inversely proportional to the seventh power of distance: $F \propto {c \over r^7} + d$ – SF. Nov 13 '17 at 14:03
  • I guess usage varies by field. I can't say I've encountered a situation where proportional was construed to simply mean monotonic. – agentp Nov 14 '17 at 11:10
  • @agentp: For continuous domain functions will usually be a bit more restricted - I'd say, infinitely differentiable and all derivatives monotonic (weakly, and not necessarily the same way as the function). Although proportionality may be defined in domains that are discrete, for specific "extracts" of the domains (vectors change proportionally if their lengths do), or for domains that are very hard to put in mathematical terms (punishment should be proportional to crime). – SF. Nov 20 '17 at 18:23