Re: [Phys-l] explanatory and response variables (was calibration )

[John Denker <jsd@av8n.com>]

On 08/09/2007 07:46 AM, Joseph Bellina wrote:
> It seems to me that graphs are basically tools to use to solve a  
> problem we are faced with, so Hugh's suggestion make sense. 

Agreed.

>  It does  
> mean of course that you have to agree on other conventions, as in  
> this case, that the experiment results will be discussed in terms of  
> cm/sec, rather than sec/cm.  Or would that be ok to use sec/cm?

I would say we do not need to agree in advance;  we can choose
sec/cm or cm/sec depending on circumstances.

This involves two related points:

 1a)  dx/dt is a derivative.
 1b)  dt/dx is a derivative.

  ... and we can calculate either one, or both, subject to mild 
  restrictions.  The definition is symmetric.  Even the restrictions 
  are symmetric.

 2) The choice of dx/dt and/or dt/dx is _independent_ of whether
  x is plotted vertically (as in middle school) or horizontally
  (as in spacetime diagrams).


I realize that if you speak in terms of _slope_, that conventionally
means rise over run, and that does depend on what is horizontal and
what is vertical.  However, if you speak in terms of _derivative_,
then everything is symmetric.

As the saying goes, you should keep an open mind, but not so open
that your brains fall out.  In this case, it is safe and indeed
advantageous to keep our minds open and keep our options open.  
The physics doesn't require doing it one way or another.  There 
is no single "natural" way to do it.  Any restrictions are a matter 
of taste, or at most a matter of convention.  And the conventions 
are context-dependent, not universal (as the example of spacetime 
diagrams shows).

Open-mindedness is particularly advantageous when studying differential
equations, where sometimes you are given dx/dt and sometimes you are 
given dt/dx.

=======

By the way, the same symmetry applies to integrals, not just derivatives.
  a) You can integrate x dt.
  b) You can integrate t dx.

... subject to mild restrictions.  Hint:  integration by parts.

Again, open-mindedness is advantageous.  Integrating by parts is 
a widely-useful technique.