Sunday, 5 March 2017

calculus - Integrating equations relating infinitely small changes

I've been studying classical mechanics recently and I have two (related) questions on calculus:





  1. In the first chapter of R. Shankar's book Fundamentals of Physics, he derives vdv=adx (in a time interval [t,t+dt] as dt0) from the definitions of a and v. Then he writes:



    v2v1vdv=ax2x1dx



    for the situation when v and x change in a time interval [t1,t2], and I don't quite understand how that follows (as the quantities are equal in the same time intervals, while the integration is with respect to v and x; and v as a function of v is completely different than v as a function of t).


  2. How to justify the variable change done here? It probably flows from the answer to the first question, but maybe there's something more to be said.


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