6
$\begingroup$

Mathematically speaking, what does it mean to say that a physical quantity is some numerical value with a “dimension” associated with it? When we say that the velocity of light is some constant, c meters per second, at first thought, it seems that we are talking about a ratio of differentials, $v=dx/dt$. But what about the "dimension" of angular momentum, ${mass} \times {length}^2/{time}$? I've never seen a differential of mass in a total derivative... $L=dmdx^2/dt$ ! Or what about the "dimension" of electrical resistance, ${time}/{length}/{permittivity}$? I've never seen a differential of permittivity, either. So, the idea that a "dimension" might be just a total derivative just doesn't seem to make sense, because the number of differentials in the numerator and denominator is not always equal.

So, what is the mathematical nature of this beast we call a "dimension"?

  • 3
    "I've never seen a derivative of mass!" - one considers the derivative of mass with respect to time in Newton's second law when the accelerating object loses or gains mass as it moves, e.g. rockets.2011-09-04
  • 1
    $dx/dt$ is a derivative, not a ratio of derivatives.2011-09-04
  • 0
    "the number of derivatives in the numerator and dnominator is not always equal." - huh? What did you have in mind?2011-09-04
  • 0
    Why do you want to associate dimensions with derivatives (or actually differentials) in the first place? Are you generalizing from the fact that velocity is a derivative? If so, why? The generalization plainly doesn’t work (and makes no sense).2011-09-04
  • 1
    @J.M.: Koilon appears to be using *derivative* for *differential* and somehow trying to equate dimensions with differentials. Thus, in the case of angular momentum he has three in the numerator and only one in the denominator, which (he notes) makes no sense as any kind of derivative.2011-09-04
  • 0
    Brian, the point I was trying to make is that the idea that "dimensions" were, okay, differentials did not work out - hence my question.2011-09-04
  • 1
    I don't see that anyone has really addressed my question, yet. What is the mathematical nature of this beast called a "dimension"?2011-09-04
  • 2
    This may not completely address your question, but you may be interested in [my answer here on Physics.se](http://physics.stackexchange.com/questions/13060/what-is-the-logarithm-of-a-kilometer-is-it-a-dimensionless-number/13087#13087).2011-09-04
  • 0
    Willie, that was interesting. A graded algbra is a new concept for me. I'll have to bone up on that one. The fact that the exterior algebra is a graded algebra suggests to me that there may be something relevant to my question here. In fact, this discussion has also stimulated my own thinking...maybe a "dimension" is like a "type" in the typed lambda calculus and the algebra(?) of typed combinators.2011-09-04
  • 0
    Koilon: In order for @Willie to be notified of your comment, you should add an `@` twitter-style in front of his name (not necessary now because he was notified by the present comment).2011-09-04

3 Answers 3