From: hrubin@stat.purdue.edu   
      
   In article ,   
   Henry Spencer  wrote:   
   >In article ,   
   >Makhno  wrote:   
   >>Is this a valid step here? The increase in kinetic energy when an   
   >>infinitesimal mass dm is accelerated out of the engine is clearly   
   >>dE=0.5*dm*c^2. This takes place over a time period T, which is an unknown   
   >>constant.   
      
   >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.   
      
   >To which I say:  if it was good enough for Leibniz, it's good enough for   
   >me. :-) The dirty little secret of calculus is that just casually treating   
   >dt like an ordinary variable gets you the right answer, every time... and   
   >since the development of "nonstandard analysis", we know why.  You can   
   >build a mathematically-rigorous non-standard number system in which dm/dt   
   >*is* a quotient and dt *is* an ordinary variable, and it's provably   
   >equivalent to the vastly more complicated epsilon-delta circumlocutions   
   >that later mathematicians invented to explain why Leibniz's calculus   
   >perversely insisted on working so well.  The engineers' deplorable habit   
   >of just dividing dm by dt and getting dm/dt turns out to be rigorously   
   >justifiable after all.   
      
   In the writings of Leibniz, it may seem to have been treated   
   that way, and if you know what you are doing and when to do   
   it, it will give you the right answer in those cases.  Even   
   in non-standard analysis, it does not always work.  If one   
   has a function of more than one variable, one can still work   
   with differentials, but not by dividing.   
      
   One can make "epsilon-delta" more understandable, and it is   
   needed to show why non-standard analysis works when it does,   
   and why.  There are other ways of working correctly with   
   differentials, and I do this often.   
   --   
   This address is for information only.  I do not claim that these views   
   are those of the Statistics Department or of Purdue University.   
   Herman Rubin, Department of Statistics, Purdue University   
   hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558   
      
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