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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?

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  • $\begingroup$ physics uses non-rigorous but intuitive math tools, which of course could be wrong. In most (or at least many cases though) they are later formalized by mathematicians to provide a rigorous foundation. $\endgroup$
    – user83548
    Commented Jun 4, 2016 at 5:06
  • $\begingroup$ If you search Wikipedia for "forms" and "Relativity", you will see how the use and interpretation of infinitesimals is now used in modern physics. $\endgroup$
    – user108787
    Commented Jun 4, 2016 at 5:15
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    $\begingroup$ Actually, we never really use infinitesimal quantities. We just pretend that we do. What we mean by those is that if someone decided to make a measurement, they could find a small enough, but finite, measurement interval on which the differential relationship is "good enough". Calculus is merely the mathematical approximation for these finite differences that is highly convenient to work with. In reality everything that starts with a $d$ in physics is really just a small enough $\Delta$. When you see mathematicians who are trying to formalize physics, you are seeing wasted effort. $\endgroup$
    – CuriousOne
    Commented Jun 4, 2016 at 5:21
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    $\begingroup$ Related: physics.stackexchange.com/q/92925/2451 , physics.stackexchange.com/q/70376/2451 , physics.stackexchange.com/q/65724/2451 , physics.stackexchange.com/q/153791/2451 and links therein. $\endgroup$
    – Qmechanic ♦
    Commented Jun 4, 2016 at 6:09
  • $\begingroup$ @Nesar Just curious, what do you mean by "non-standard analysis"? Asking because the term is usually associated with Robinson's "non-standard analysis", en.wikipedia.org/wiki/Non-standard_analysis. $\endgroup$
    – udrv
    Commented Jun 4, 2016 at 7:48

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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.

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