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As far as I know, in physics, calculus is approached using nonstandard analysis in which $dx$, $dy$, etc. (infinitesimals) are treated as fixed, extremely small quantities rather than the standard analysis approach using limits where they are treated as something that approaches $0$.

I understand that the nonstandard approach is very intuitive and easy to understand. In fact, I had been doing calculus via the nonstandard approach till a few days ago when I encountered the philosophical questions related to infinitesimals. I got extremely confused due to this approach and when I approached calculus using limits, I felt like I had gained a whole new level of understanding and conceptual clarity.

Now, in physics, for a function $f$, $f'(x)$ or $\dfrac{df}{dx}$ is interpreted as the rate of change of $f(x)$ with a very small change in $x$, i.e. $dx$. It is also interpreted as approximately the slope of tangent to the curve of $f$ at $(x,f(x))$. This approach and geometric intuition is also used to derive the fundamental theorem of calculus which states that if $F(a)$ gives the area under the curve of $f(x)$ from $x = 0$ to $x = a$, i.e. $$F(a) = \int_0^a f(x)dx$$ Then, $$\int_a^bf(x)dx = \int_0^bf(x)dx - \int_0^af(x)dx = F(b) - F(a) = F(x)\Bigg|_a^b$$ where : $$F'(x) \text{ or } \dfrac{dF}{dx} = f(x)$$ The nonstandard approach is also used to derive certain formulae like that of work which is derived as follows:

  • For an infinitesimal displacement $dx$, the infinitesimal work done i.e. $dW$ is $F_2(x)\cdot dx$
  • The total amount of work done, i.e. $W$, is $\int_a^bF_2(x)\cdot dx$ (Note: here, $F_2(x)$ denotes the force experienced by the particle at position $(x)$. For example, if we are talking about electrostatic force, $F_2(x) = \frac{q_1q_2}{4\pi\varepsilon_0x^2}$.)

So, basically, most of calculus used in physics is approached using infinitesimals and nonstandard analysis.

But, standard analysis seems much more rigorous to me and makes much more sense. I have asked a few friends who asked their teachers how one can use standard analysis in physics instead of nonstandard analysis but none of the teachers seemed to bother.

So, I'd like to know how I can approach physics via standard analysis.

PS: I'm currently in 10th grade and have only covered the basics of 11th grade yet. An answer that I can comprehend with not much knowledge about advanced mathematics would be appreciated.

Edit : I am very grateful for the two answers that I have already received. I recklessly assumed that nonstandard analysis and the heuristic use of infinitesimals are one and the same which is not the case, as pointed out by users Qmechanic and PM 2Ring. I would like to clarify that whenever I used the term 'nonstandard analysis', I was actually referring to the treatment of $dy$, $dx$, etc. as actual, very small numbers and of $\dfrac{dy}{dx}$ as a ratio...

Rajdeep Sindhu
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  • A recent related question: https://physics.stackexchange.com/q/572956/123208 – PM 2Ring Aug 16 '20 at 03:36
  • Possible duplicates: How to treat differentials and infinitesimals? , Rigorous underpinnings of infinitesimals in physics and links therein. – Qmechanic Aug 16 '20 at 04:12
  • Many, if not most, problems in physics (and elsewhere) are solved using numerical methods, where treading dx, dy &c as small quantities is the way things are done. So perhaps there's a good reason for using what you call a "nonstandard" approach. – jamesqf Aug 16 '20 at 05:00
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    OP seems to conflate non-standard analysis and common heuristic treatment of infinitesimals. – Qmechanic Aug 16 '20 at 08:28
  • @Qmechanic Wait, they're different? I intended ask "How can I approach calculus used in physics without interpreting $dx$, $dy$ etc. as numbers or $\dfrac{dy}{dx}$ as a ratio" through this question – Rajdeep Sindhu Aug 16 '20 at 09:08
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    @Rajdeep Yes, they're different. Non-standard analysis dates from the middle of the 20th century, but physicists have been using infinitesimals heuristically since the early days of calculus, and they even saw some usage before Newton & Leibniz, eg Fermat's adequality. – PM 2Ring Aug 16 '20 at 16:26
  • @PM2Ring Is the heuristic use of infinitesimals consistent with using limits to define derivatives, integrals etc.? And, should I edit my question to include that I am using the term 'non standard analysis' to actually refer to the heuristic use of infinitesimals and their treatment as actual numbers? I think I should... – Rajdeep Sindhu Aug 16 '20 at 18:11
  • Well, your question already has 2 answers that refer to non-standard analysis, so changing that now would invalidate those answers, to an extent. But I guess it's ok for you to clarify that you really meant to refer to the heuristic use of infinitesimals. – PM 2Ring Aug 16 '20 at 18:22
  • I'm a bit unclear as to what you're looking for. If you don't want to work with infinitesimals, then simply skip the definition of $dW$ and directly define the work done along a path to be $\int F(x) dx$. No infinitesimals needed. The definitions of derivatives and integrals can be given in terms of limits, so what else do you want? – J. Murray Aug 16 '20 at 19:32

2 Answers2

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So, basically, most of Calculus used in Physics is approached using infinitesimals and non standard analysis.

This premise is untrue. While physics arguments about infinitesimals may resemble typical arguments in "non-standard analysis", elementary physics usually does not operate at levels of rigor where you could clearly decide whether it is using non-standard analysis or not. Physics is not interested in foundational questions of analysis, and e.g. the derivative of a function is an approximation to its slope regardless of which foundation you're using (it may be more or less work to derive this depending on your foundation, but it's still always true).

If you are looking for rigor, there is often an equally valid interpretation of physical "infinitesimals" in terms of standard differential forms, e.g. $\mathrm{d}W = F(x)\mathrm{d}x$ is simply the definition of a 1-form called $\mathrm{d}W$, whose integral over paths $\gamma$ is defined to be the work $W[\gamma] = \int_\gamma \mathrm{d}W$ along the path.

ACuriousMind
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  • Thank You for the answer, first of all. I can't really make sense of this statement : "$dW = F(x)dx$ is imply the definition of a 1-form called $dW$, whose integral over paths $\gamma$......along the path". Being a $10^{th}$ grader with some elementary knowledge of $11^{th}$ grade, is there some way I can interpret this? – Rajdeep Sindhu Aug 16 '20 at 05:51
  • Also, when I said that standard analysis makes much more sense to me, I meant that to me, using infinitesimals gives an idea of approximating the result (the approximation being approximately very accurate) whereas using limits and the idea of approaching the the exact result gives an idea that we are actually evaluating the exact result. – Rajdeep Sindhu Aug 16 '20 at 05:59
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There is no difference in rigor between NSA and standard analysis. (In terms of model theory, they are equiconsistent.) However, most of the infinitesimal calculus you see in physics papers and books would need to be reworked or elaborated slightly to make it into NSA.

It's typically trivial to translate back and forth between the two languages. Scientists and engineers should be fluent in both.

user272406
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