Book recommendation on infinitesimals

Question

I'm a sophomore mechanical engineering student and I'm looking for a book which completely describes the concept of "infinitesimals" (used in thermodynamics or dynamics or...), In fact, I was taught that dy/dx is not quotient ,but in physics it seems they are treated and being manipulated like numbers

Answer 1

I should like to suggest:

H. Jerome Keisler, "Elementary Calculus: An Infinitessimal Approach". Download from Author's website under Creative Commons Licence

This one too is a good read and free. Three warnings:

1. It may be a little different from what you have in mind;

2. It's almost certainly different from what most physicists have in mind when manipulating differentials like numbers especially if over the age of 40 (I risk drawing flame with that comment, so I'll declare tha I'm 51);

3. It will not teach you anything that you will find "practical" to your physics endeavors, in the sense that learning about differential forms as advised in Dan Piponi's Answer will teach you material that you will use over and over again in physics;

However, what it will give you is an understanding of a rigorous framework that justifies many of the sleights of hand you are seeing in your classes. You sound like someone who wants to know a bit more about mathematics that what's needed to apply a cookbook recipe. Some background knowledge that might convince you that the hyperreals and Robinson's nonstandard analysis are worth a look (and indeed made the mathematics world sit up and take heed): Robinson himself proved that his axioms for hyperreals are consistent relative to the real numbers. This means the following: you may know that Gödel's incompleteness theorems rule out the proof of the consistency of any mathematical system containing roughly Peano's concept of integer from within the system itself. However, one can prove relative consistency: Robinson proved that if the real numbers form a consistent mathematical system, then so do the hyperreals.

The book itself is extremely readable, and given the level your question bespeaks, you will likely be able to skim many sections simply glossing them with the thought "how does this differ from the conventional approach to this topic?". I would advise the following course of action:

1. Read the Introduction;

2. Skim the the parts of Chapter 1 up to the beginning of Section 1.4 "Slope and Velocity: The Hyperreal Line" as you will likely already know all this;

3. Read Chapter 1 from Section 1.4 "Slope and Velocity: The Hyperreal Line" onwards (inclusive of Section 1.4) carefully

4. I think you will then be in a position to know whether further study of this book will be worthwhile for you.

I would describe the book as written informally but clearly by a mathematician at an undergraduate, but not belittling, level, with the informal language conveying a sketch of how the ideas can be made altogether rigorous. It is not altogether unlike Roger Penrose's "Road to Reality" in style: its style seeks to broaden the minds of the curious.

If, after mastering the notion of a hyperreal and an infinitessimal, you have an appetite for more, then look up synthetic differential geometry. Urs Schrieber in his answer to the Physics SE question "How to treat differentials and infinitesimals?" gives you an invitation and some links to this topic: it is not for the mathematically feint of heart.

Answer 2

I think you might not find a book on exactly what you want. Infinitesimals tend to get used loosely by physicists and there isn't really a rigorous theory behind how they are, in practice, used. It's something you pick up "on the job" so to speak.

Having said that, many lines of reasoning involving infinitesimals can be made rigorous in various ways.

Some arguments involving infinitesimals can be interpreted as arguments about differential forms. You can learn about these from books on differential geometry. I learnt about this stuff from Geometrical Methods of Mathematical Physics by Bernard Schutz. In that book you'll find arguments from, say, thermodynamics, that look like familiar ones using infinitesimals, but in actual fact are using differential forms. It can be challenging unless your calculus is really strong.

Another approach is via non-standard analysis. For example there's the book A Primer of Infinitesimal Analysis by John Bell. It gives a rigorous theory that allows you to use infinitesimals a lot like ordinary numbers. But this approach is called non-standard for a reason. It might not be what you really want and there's some heavy duty mathematics in the background.

But I think the most practical approach is to get lots of practice with analysis, in particular $\delta-\epsilon$ arguments as covered in just about any analysis textbook. Then you'll recognise many arguments apparently using infinitesimals as really being shorthand for rigorous proofs using limits. Remember that when you originally defined the derivative, the thing you call $dx/dy$ is (usually) defined as a limit of a quotient of finite size values $\Delta x/\Delta y$. Physicist reasoning involving infinitesimals can largely be seen as being sloppy about how you take the limit.

Or do what everyone else does. Go with the flow and treat infinitesimals just like numbers :-)

Answer 3

Since differential geometry seems to be the flavour of this thread, my recommendations are as follows.

Personally, I think the best text for differential forms is set out in R.W.R. Darling's Differential Forms and Connections: see link here.

For a physics slant, try Nakahara's Geometry, Topology and Physics, the text can be found here.

Personally, when I was a mathematics undergraduate, I found John McCleary's Geometry from a Differentiable Viewpoint (here) and Differential Geometry by Kreyszig (here) indespensible but appreciate it may be more suited to a math Major.