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I want to ask about infinitesimals and non-standard analysis. In physics we always use $\mathrm dx,~\mathrm dv,~\mathrm dt$ etc. as infinitesimal quantities. When we deduce equations in physics, when we set up integrals and in many other instances we use infinitesimals. For example in the first law of thermodynamics we have: $đQ=\mathrm dU+đW$.

Now when we ask teachers about what is $đQ$ here, they say it is an infinitesimal amount of thermal energy. So my question is, is it necessary to treat the whole non-standard analysis as an axiom for physics or do we have standard calculus explanation for physicist's use of $\mathrm dy/\mathrm dx$ as a quotient in all cases?

Qmechanic
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Nesar
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1 Answers1

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Physicists do you the rules of calculus but can be sloppy in their notation.

The symbols $\delta, \Delta$ and $d$ often seem to be used interchangeably to mean a (small or infinitesimal) change in something or better still a final value minus an initial value.

So in your equation $dU$ is the change in internal energy of a system or final internal energy minus initial internal energy $U_f-U_i$.

However with this notation the use of $dW$ and $dQ$ is often frowned on as there is no initial amount of heat and final amount of heat it is just the amount of heat leaving or entering the system and the same is true of the work done.

To indicate this instead of using $d$ a new symbol $d$ with a line/bar through it is used (I do not known how to write this but it like $\hbar$). That being said if you look through many textbooks $\delta$ and $d$ are used but often with an explanation that they represent inexact differentials.

Farcher
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