Re: [Phys-l] Symbols for Kinematics

[John Denker <jsd@av8n.com>]

On 08/11/2007 04:55 PM, Jeffrey Schnick wrote:
> Regarding
> <http://www.av8n.com/physics/intro-acceleration.htm>
> In light of the message in section 2.2 of that document that v is not a
> function of t,

On that page, I try to be very precise and consistent.

We get into trouble if you try to mix the precise terminology with
everyday sloppy shorthand terminology.

There is a function that gives us v as a function of t.
We call this function G.  This function is not properly called v.

I just now added more discussion of this point, near the end of
the "functions" section.
  http://www.av8n.com/physics/intro-acceleration.htm#sec-func

> I need some clarification on the meaning of definition 1:
> a:=dv/dt
> I think it implies that v is a function of t because a function is what
> one takes the derivative of.

Well, there are two options.  

  1) If you insist on taking derivatives of functions only, you
   can say that acceleration is the derivative of the G function.
   This changes the notation in the definition of acceleration
   to something like:
        a := G' (t)
where
        v = G(t)

  2) I prefer to take a more basic approach, a more geometric
   approach, a more physical approach.  For me, the derivative
   can be defined as the limit of the difference quotient:
               v2 - v1
          lim ---------                [1]
               t2 - t1

   with the understanding that v2 and t2 go together, and v1
   and t1 go together.  I see no reason to require that the
   numerator-variables be known as a function of the denominator-
   variables;  equation [1] works just as well the other way
   around, and it also works if both of them are known as
   functions of some third "parametric" variable.

   The physical picture here is that (t,v) is some point in an
   abstract space, and two such points, in the limit that they 
   are near each other, define a tangent vector in the tangent
   space.

   Forsooth, if we are going to pick apart the derivative
   into independent differentials, as I do in the very next
   line of my derivation:
     http://www.av8n.com/physics/intro-acceleration.htm#eq-dv-adt
   then we simply must have a broader view of things;  we
   cannot simply regard (d/d) as just a machine for taking
   derivatives of functions.

   This broader view pays off handsomely in thermodynamics,
   because trying to keep track of what is an "independent"
   variable versus "dependent" variable is somewhere between
   a bad idea and an impossibility.

   The broad view can be conveniently and fruitfully formalized
   in terms of differential forms.
     http://www.av8n.com/physics/thermo-forms.htm

   
   *) I don't expect high-school students to know about differential
    forms.  They can be content writing finite differences: Δt, Δx,
    Δv, et cetera.  They can start on line 3 of my derivation.

   *) On the other hand, it is useful for teachers to know that
    there is a well-behaved formalism for talking about dt, dx,
    dv, et cetera even when they are not part of the stylized
    (d/d) derivative notation.

    For more on this (probably more than you wanted to know, unless
    you are doing some serious thermodynamics) see:
      http://www.av8n.com/physics/partial-derivative.htm