From: nebusj@rpi.edu   
      
   henry@spsystems.net (Henry Spencer) writes:   
      
   >No, it takes place over an infinitesimal time dt.  So just divide both   
   >sides by dt, and you get dE/dt = 0.5*(dm/dt)*c^2 as suggested.   
      
   >Any mathematicians in the audience are probably having apoplexy at the   
   >"just divide by dt" part, and will tell you at great length that something   
   >like dm/dt is not really a quotient and you can't just casually treat dt   
   >as if it were an ordinary variable.   
      
   	Henry, I'm afraid I don't know any mathematicians who'd   
   protest the manipulation of dt like you've used in this context.   
   Granted I got the bulk of my training at RPI, where the mathematics   
   program offers the ``Applied Mathematics'' and the ``More Applied   
   Mathematics'' tracks, but poking through the references I have on   
   my desk shows that sort of move done without complaint, past a rare   
   note that proving this rigorously requires analysis not included.   
      
   	The point of emphasizing to students that dE/dt is not a   
   fraction is that if students believe it *is* a fraction, they'll   
   quite happily cancel the d's out of both sides and get an answer   
   that's not even wrong.  Even with the emphasis to not think about   
   this like a fraction that happens disappointingly often.  Making   
   the point that it's not a fraction reduces the risk -- and it has   
   the benefit of getting students to think of the derivative as this   
   thing done to a function, not just a quotient, which is quite a   
   useful point of view.   
      
   	Also the sharper (or more troublesome) students will point   
   out that if you let dt go to zero, then you get a zero over zero form   
   and how do you get any answer from that ... which requires appealing   
   to all sorts of explanation that won't fit in any non-major's courses.   
   Best to try avoiding the issue altogether.   
      
   --   
   								Joseph Nebus   
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