Re: [Phys-l] Motion in 1D, vectors and vector components

[John Denker <jsd@av8n.com>]

On 08/15/2007 12:02 PM, Jeffrey Schnick wrote:

> I think the word "projection" is, in general, used for both what you
> call a component and what you call a projection.  

Actually it seems to me the word "component" is more overloaded
than the word "projection".

> For a couple of
> examples where it is used to mean the number-with-units see:
> <https://carnot.physics.buffalo.edu/archives/2002/09_2002/msg00392.html>
>
> where it says:
> " 'component' = as Bob says below, the projection of the vector of
> interest onto the axes, what is traditionally labeled A_x and A_y
> (signed scalar quantities)."
> or see:
> <http://www.av8n.com/physics/acceleration.htm> 
> where it says:
> "The scalar acceleration can be considered one component of the vector
> acceleration, namely the projection in the 'forward' direction (although
> this is undefined if the object is at rest)."

Point taken.

I hereby change my mind about this.  It appears that the meaning
of "component" is context dependent as follows:
 -- A component of a BLoC is a /matrix element/ which is a number
 -- A component of a geometrical vector is /projection/ which is
   a vector unto itself.

Note that the term /matrix element/ is well established and 
unambiguous in the mathematics literature, and the physics
majors will encounter it again when they get to quantum
mechanics.  Also /projection/ has an unambiguous hands-on
meaning in terms of shadows and projective geometry.  So
it seems pretty clear to me that all we really need to do
is disambiguate the term "component".

> Thomas Moore in his book /Six Ideas That Shaped Physics/ uses the
> terminology "vector component" for the value-with-units and "component
> vector" for what you call a projection.

That's an interesting suggestion, but too ugly and confusing.   I don't 
think I could keep that straight.  I especially don't think students 
could keep that straight.

> Perhaps the use of the following terms would help avoid confusion:
>    vector component value
>    component vector
>
> Note: I hesitate to use the word "scalar" for the vector component
> value.  I don't think of a vector component value as something that
> transforms as a scalar (remains the same), for instance under rotation.

Agreed!  

I think saying "matrix element" rather than "scalar" avoids this
problem.

> "What is the derivative of a scalar field, say dT/dx?  Is it a scalar,
> or a vector, or what?  It is neither a scalar nor a vector, as you can
> easily appreciate, because if we took a different x-axis, dT/dx would
> certainly be different."

The smart thing to do in such a case is to ask about the gradient,
namely dT, rather than the directional derivative dT/dx.  It turns
out that dT is a vector ... in particular, a one-form (as opposed
to a pointy vector).  This is discussed at
  http://www.av8n.com/physics/thermo-forms.htm
This is so simple, practical, and reliable that IMHO it would be 
sheer masochism to do thermodynamics any other way.