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: [Phys-l] Types of scalars



I think that in discussing this topic, we have to agree on some conventional
definition of the notion of scalar. In one such definition a "scalar" is synonym
of an "invariant". If this is accepted, then it is relatively straightforward to
indicate types of scalars by the kind of transformation under which they remain
invariant. But this taxonomy is rather fuzzy since a characteristics which is a
scalar under one kind of transformation may be not a scalar under another
transformation. Here are some familiar examples:

1) a vector component - a scalar under translations, not a scalar under
rotations or inversions (reflections)
2) time, energy, distance - scalars under 3-rotations, not scalars under
4-rotations
3) proper time (norm of a 4-displacement), rest mass (norm of a 4-momentum) -
scalars under 3- and 4-rotations and inversions
4) work (F*S) - a dot (scalar) product of force and spatial displacement -
scalar under 3-rotations and inversions, not a scalar under 4-rotations
5) charge - a scalar under all known transformations
6) parity - a scalar under inversions (when even, P = 1), not scalar under
inversions, when odd (P = --1),
etc.

Moses Fayngold,
NJIT



________________________________
From: Scott Hill <shill@wso.williams.edu>
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Sent: Sun, January 2, 2011 12:37:30 AM
Subject: [Phys-l] Types of scalars

I've been reflecting on fundamental mathematics that I gloss over in
my introductory physics classes, and it occurred to me that there are
actually three types of scalars that we use in introductory physics:
1) Never-negative scalars, like mass or length or distance.
2) Scalars which can be negative, and the zero is significant: like
charge or flux or current, or changes in temperature, or vector
components.
3) Scalars which can be negative, and the zero is not significant:
like temperature (in Celsius) and potential energy.

I am thinking it would be convenient to make this distinction to my
introductory students, because they behave in different ways (for
example, Category-3 scalars are never multiplied). Has anyone seen
these distinctions made before? I think all physicists are aware of
these categories, but I don't know that anyone ever explained them to
me.

More importantly, are there NAMES for these three categories?

They can be broken up even further if one has a mind:
1a) Quantities like mass or length or duration, which can be added and
subtracted directly.
1b) Vector magnitudes like speed, which probably can't be added
without going back to the original vector.
2a) Fluxes like current or work: the sign indicates direction of flow.
2b) Surpluses, like charge (or bank accounts): the sign indicates
presence or absence.
2c) Changes in quantities, like ∆T: the sign indicates increase or decrease.
3) I can't think of any subcategories of this one.

Too many categories may be more confusing than helpful, however.


/
:@-) Scott
\
_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l