Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: dimensionless units



At 01:32 AM 12/1/00 -0500, David Bowman wrote:
what is described below is not a serious proposal (in case anyone was
wondering).

It is clever enough to deserve at least a little bit of serious consideration.

My proposal for including 'angle' as a base dimension is to arbitrarily
consider the Levi-Civita [epsilon] symbol to have the dimension of
inverse 'angle', (A^-1)

1) In 3D, there are _two_ ways in which angles are related to vectors:
a) As David mentioned, there is the cross product, |A| |B| sin theta,
which can be written in terms of the Levi-Civita symbol epsilon_i_j_k.
b) There is also the dot product, |A| |B| cos theta, which can be
written in terms of the Kronecker symbol delta_i_j.

It seems to me that attaching dimensions to the sines will not be
sufficient; one also needs to go after the cosines.

2) The Levi-Civita symbol can be used for things other than cross
products. For instance, we have the following mathematical identity:
epsilon_i_m_n epsilon_j_m_n = 2 delta_i_j
where we are invoking the Einstein summation convention for repeated indices.

Now it's pretty clear that we need to have a Kronecker delta that is
dimensionless. Therefore David's dimensionful epsilon symbol cannot
possibly be the whole story; either it coexists with the usual
dimensionless epsilon symbol, or possibly it coexists with something with
reciprocal dimensions.

In either case, this places a burden on the user to decide which version to
use.

3) I think it should be possible to fix up the bug mentioned above. But it
won't be automatic. It will involve checking all the formulas of physics
to see which are covariant, which are contravariant, and which are
invariant with respect to a change in what units (radians .. cycles ..
degrees ...) are used to measure angles.

On the third hand, that wouldn't be a bad thing to do!

Here's a start:

a) The formula for centrifugal field
A = R omega^2
should be replaced by
A = R omega^2 radian^-2

b) The formula for the number of times (S) a bell is struck by a crank
rotating at given frequency (f) during a given period (t)
S = f t
should be replaced by
S = f t cycle^-1


===========================================

A proverb that may help students:

"Dimensionless is not the same as meaningless."

I think the "percent" example is a good illustration. "100%" is synonymous
with unity. You can multiply by 100% any time you feel like it. It's a
freebie. But "%" by itself is not a freebie.

"Radian" is not a freebie, unless you are absolutely sure that all the
formulas you care about are using radian measure -- which in the real world
is not a safe assumption.

The foregoing applies to equality relationships. If OTOH you are doing
dimensional analysis, or other proportionality relationships, then % and
all other dimensionless critters are freebies.