How does mathematical coefficients differ from "physical coefficients"?

Question

In the Talk page of Wikipedia Coefficient I read this comment:

As far as I can tell, the mathematical definition should imply that coefficients are unitless, however, the physical sciences have been using "coefficient" for factors that include dimensions for a long time, where "constant" or "factor" would be a more informative term.

In the same page, examples of "physical coefficients" have both dimensionless and dimensionful coefficients. I am trying to understand the difference of how "coefficient" is used in mathematics and physics.

Thanks.

Answer 1

The comment doesn't mean that physical coefficients must have a unit; it just means that some do.

There is no real difference, as far as I know, in the definition of coefficient for math or physics. The difference is that, in mathematics, units are rarely assigned to the quantities used, whereas in physics, units are fundamentally important to keep track of. This complicates the description of equations with coefficients slightly; consider, for instance, the kinematics equation

$$r = \frac{1}{2} a t^2 + v_0 t + r_0$$

To a mathematician, this equation describes $r$ as a simple quadratic in $t$, given a few other parameters, and we don't have to do any verification up front that it makes any sense; it is a certainly a proper family of functions. To a physicist, this equation can only make sense after we have verified that the units of the terms being added and equated match up. This means that the coefficient of each term must adjust to ensure that the product has the same unit as all of the other terms, as they do, in this case.

Answer 2

Like Dustan has stated, there isn't a significant difference between them. It's more of a subtlety, if anything, that separates them.

In physics, coefficients refer to actual physical things. In mathematics, it's not like that. Mathematics tends to characterize general abstractions. For example, in the quadratic equation $ax^2+bx+c=0$, there is nothing physical about $a$,$b$, and $c$. They don't represent any physical thing. However, in $F_{\text{friction}}=\mu N$, $\mu$ represents a very real thing: The coefficient of friction for a particular substance.

Perhaps an equivalent way of looking at this illustrates things better: Physical coefficients are applied mathematical coefficients. For example, $ax^2+bx+c=0$ is a quadratic equation that can become the equation for distance: $d=d_0+v_0t+\frac{1}{2}at^2$. This is done by noticing that: $$x=t, a=\frac{1}{2}a, \, b=v_0, \, c=d_0-d$$

In this sense, the quadratic's coefficients now have physical meanings. Before, they were simply symbols of abstraction. A simpler example would be the equation for linear thermal expansion: $$L=L_0+L_0\alpha \Delta T$$ This is simply a linear equation of the form $y=mx+b$ where

$$y=L, \, x=L_0, m=(1+\alpha \Delta T), b=0 $$

Just like before, the coefficients now take on a meaning, rather than just being an abstraction. Also, note that the coefficients--like in the abstraction--are things we know whilst the $x$ is something we don't know. (In most cases of this simple problem, $\alpha$ and $\Delta T$ are known quantities.)

Yet another example would be:

$$v=v_0+at$$

where this equation takes the form of, again, $y=mx+b$ with

$$m=a,\, b=v_0,\, x=t$$

$a$, acceleration, is usually taken to be constant and the intial velocity, $v_0$, is always constant. I hope this has illustrated the point better. :P

(Some of my remarks here are debatable because these equations are not the more complex and Calculus-esqe equations that describe the same thing, so take them a bit lightly. There are assumptions in all of these equations, I think.)

Is there some kind of list of coefficients according to what they do?

There is this: http://en.wikipedia.org/wiki/Physical_constant

Constants are just coefficients that don't vary and occur a lot. For example, acceleration due to gravity (on the Earth) is $9.8$ m/s$^2$ and is a particular value of the coefficient $a$ (when $a$ is taken to mean acceleration, of course). Here is another good reference: http://en.wikipedia.org/wiki/Variables_commonly_used_in_physics

I also use this a lot to understand some of the basic facts of the variables in Physics, a lot of them being coefficients: http://en.wikipedia.org/wiki/List_of_physical_quantities