What is a good book for learning rigorous calculus?

[Rib5]

I'm not sure if the title correctly says what I am looking for. I'm a few years out of college and I'm trying to review some electromagnetics topics. A lot of the "proofs" in my EM book seem to take a lot of shortcuts, or use "intuition" to explain why some calculus operation can be simplified. There is a lot of treating infinitesimals (i.e., dx, dy, dz) as variables.

I always heard that engineers and physicists doing math will drive mathematicians crazy; I am an engineer myself, but doing something without fully understanding the justifications is something that always made me uncomfortable.

I tried going back to my college level calculus textbooks, but I noticed that even those have some "you can do this here, just trust us..." type explanations. To make matters worse, it seems like every website, every engineering textbook, and every math textbook use different notation and meanings for calculus things, making it hard for me to follow.

Is there any calculus textbook book that someone can recommend that they feel really gave them a really good, rigorous, understanding of the material, without using shortcuts like separable variables in ODE, etc, but actually doing the math the "right" way, even if it is less intuitive?

 

[micromass]

Yes, there are definitely books out there which treat calculus rigorously. As you might expect, those books will be very heavy on proofs. So you might first want to think about brushing up on proofs.

In any case,I highly recommend Nitecki's calculus decontructed. It treats everything rigorously and is meant as a second course. It's not easy though!

Feel free to PM me if you want more help!

 

[Rib5]

Yes, there are definitely books out there which treat calculus rigorously. As you might expect, those books will be very heavy on proofs. So you might first want to think about brushing up on proofs.

In any case,I highly recommend Nitecki's calculus decontructed. It treats everything rigorously and is meant as a second course. It's not easy though!

Feel free to PM me if you want more help!

Thanks for the reply! Would you say that these books helped you in not only understanding calculus, but also confidence in using calculus? I guess my end goal is confidence in using calculus for solving difficult EM problems, and possibly understanding some of the proofs for EM theorems.

 

[micromass]

Thanks for the reply! Would you say that these books helped you in not only understanding calculus, but also confidence in using calculus? I guess my end goal is confidence in using calculus for solving difficult EM problems, and possibly understanding some of the proofs for EM theorems.

Can you be somewhat more clear? I understand that you already "know" some calculus, so you know how to use it. What in terms of "using calculus" would you say are you not comfortable with now?

 

[Rib5]

Can you be somewhat more clear? I understand that you already "know" some calculus, so you know how to use it. What in terms of "using calculus" would you say are you not comfortable with now?

Right now I'm trying to follow some of the proofs for the formula of divergence in Cartesian coordinates. So I'm trying to follow the proof that shows

$\lim_{v\to o} \frac{\oint _S \vec A \cdot d \vec S}{v} = \oint _v \nabla \cdot \vec A dv$

Most of the proofs use a cube, and then find the flux through each side (e.g., top, bottom, right, etc.). In one of the steps, my book uses Taylor series and shows that the flux through one side of the cube is:

$\iint \vec A \cdot d \vec S = dy \, dz \, A_x (x_o, y_o, z_o) + \frac{dx}{2} \frac{\partial A_x}{\partial x} |_P + higher\ order\ terms$

where $P(x_o, y_o, z_o)$ is a point in the middle of the cube. So, first of all, I am confused about how on the right side you can have an equation that just contains differentials like this. I thought differentials didn't have any meaning on their own, and were really just part of the notation for derivatives and integrals. The proof goes on to combine the flux through all sides, and gets an equation with dx dy dz, and then just combines them into Δv, which, intuitively makes sense... Then it cancels out the Δv in the limit.

Another thing is, why can you combine dx dy dz into Δv, and not dv?

Lastly, in one of the further steps it says you can get rid of the higher order terms because "the higher-order terms will vanish as Δv→0." That statement makes me a little uneasy as well.

These things might be completely right to do, but it just seems fishy to me, and I'm not totally following. My thought process was that I need to understand calculus more rigorously to understand why you can do things like this. Maybe I just need to re-read my regular calculus textbook and try to understand the material in greater depth than I previously did.

 

[micromass]

Yes, they should make you feel uneasy. I know this is how physicsts work and all and it apparently works out for them. But some people desire a bit more rigor.

In any case, now I know a bit more of your issues, I'm going to recommend two books:

1) Keisler's free book: https://www.math.wisc.edu/~keisler/calc.html
What will this book teach you? This book will teach you what infinitesimals are rigorously. They will allow you to work with $dx$ rigorously on its own. This might help you a lot to make rigorous sense of what you're talking about in the previous post. It's a great book. Very rigorous. Only thing he doesn't do is actually construct the system of hyperreal numbers. But that's quite tricky to do.

2) Hubbard and Hubbard "Vector calculus, linear algebra and differential forms" http://matrixeditions.com/5thUnifiedApproach.html (later editions are better)
This deals a bit with electromagnetism already and shows how to put it on rigorous footing. It deals with expressions like $dx$ on its own too, but gives it a different (but related) interpretation than Keisler, namely that of a differential form.

 

[Rib5]

Yes, they should make you feel uneasy. I know this is how physicsts work and all and it apparently works out for them. But some people desire a bit more rigor.

In any case, now I know a bit more of your issues, I'm going to recommend two books:

1) Keisler's free book: https://www.math.wisc.edu/~keisler/calc.html
What will this book teach you? This book will teach you what infinitesimals are rigorously. They will allow you to work with $dx$ rigorously on its own. This might help you a lot to make rigorous sense of what you're talking about in the previous post. It's a great book. Very rigorous. Only thing he doesn't do is actually construct the system of hyperreal numbers. But that's quite tricky to do.

2) Hubbard and Hubbard "Vector calculus, linear algebra and differential forms" http://matrixeditions.com/5thUnifiedApproach.html (later editions are better)
This deals a bit with electromagnetism already and shows how to put it on rigorous footing. It deals with expressions like $dx$ on its own too, but gives it a different (but related) interpretation than Keisler, namely that of a differential form.

Great, thanks a lot for your help! I really appreciate that you tried to understand exactly what it was I needed. I was having a hard time putting into words exactly what my issues were. I will take a look at those two books.

 

[MidgetDwarf]

Courant Calculus.

 

[bacte2013]

Hubbard/Hubbard, Marsden/Tromba, and Zorich. I see that Professor Micromass already recommended H/H. The M/T book is not theoretical, but it has many interesting problems about applications in physics, which I believe is one of your interest. The two-volume set of Zorich rigorously treat the single- and multi-variable calculus with a huge emphasis on motivation and intuition based on the physics.

 

[NathanaelNolk]

I'm going to add to what micromass said earlier, you should definitely go with Keisler's book. You'll have a deeper understanding of differentials and their algebraic manipulations (such as the equation you posted earlier) will make more sense.