[boost] Re: Dimensional Analysis Interest?

From: Deane Yang (deane_yang_at_[hidden])
Date: 2003-10-16 13:23:10

First, I agree with what Jan Langer wrote in his message about the 
design of the library.

Jan Langer wrote:

> i think you confuse something here: quantities and units (maybe i am 
> confusing the name ;-))

I am under the impression that the standard terminology is "dimensions"
(refers to things like mass, length, and time without any particular 
units) and "units" (things like meters, grams, seconds).

>  in an abstract sense this should be 
> like points in an n-dimensional space.

(I apologize for behaving like a pedantic mathematician here going way 
off-topic, but I've always wondered what the right abstract description 
of dimensional analysis was and I'm very happy to have figured it out.)

For what it's worth, although your description of what a dimension
library should be is quite good, it's not really points in an 
n-dimensional space.

Here is a precise mathematical description of dimensional analysis. The 
thing to notice about dimensional analysis is that all of the things 
being measured are 1-dimensional. Doing dimensional analysis properly 
for higher dimensions (for example, dealing with position in 3-space and 
force vectors) is somewhat more complicated (I do sometimes dream of a 
library that
properly joins dimensional analysis with linear algebra.).

Each basic dimension corresponds to a different 1-dimensional real 
vector space (i.e., the real line without the co-ordinate labeling). 
Each possible "derived" dimension (which consists of a product of 
fundamental dimensions raised to integer or even rational powers) 
corresponds to
a tensor product of the appropriate number of the basic 1-dimensional 
vector spaces or their duals (if the corresponding exponent is 
negative). Rational powers are a little harder to describe, but also 
have a natural mathematical description.

Defining units for a given dimension corresponds to choosing a basis
for the corresponding 1-dimensional vector space (all you need to do
is choose a single vector and declare it to be a unit vector). A fancy
way to say this is to choose an invertible linear map from the vector 
space to the real line. You can do this for any of the dimensions, 
whether they are fundamental or derived. However, if you choose units 
for each of the fundamental dimensions, then they naturally induce units 
for each of the derived dimensions. Mathematically, this corresponds to 
saying that if you choose linear maps from each fundamental vector space 
to the real line, then there is a naturally induced linear maps from a 
derived vector space (i.e., a tensor product involving the fundamental 
vector spaces and their duals) to the real line.

None of this helps anyone do anything; nobody really needs to know any 
of this. But one way to distinguish between dimensions and units is that
dimensions corresponds to doing abstract linear algebra and units 
corresponds to doing linear algebra with respect to fixed bases.
Changing units (like from meters to feet) corresponds to changing the 
basis of the corresponding vector space.