# Calculus Calculus 10th Ed by Salas enough for theoretical physics?

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1. Mar 13, 2015

### shinobi20

I am currently reading Calculus Volume 1 by Tom Apostol, I find it very good but hard. I often get stuck and it took me very long time to move from page to page (mostly pages with theorems). I am done reading till half of chapter 2. I just want to know if reading apostol is really a MUST for theoretical physics (NOT going to string theory) or can I resort to Calculus by Salas which I think is rigorous although not as rigorous as apostol. Any suggestions?

2. Mar 13, 2015

### DrewD

In my opinion, there is no reason to start with a very rigorous calculus book for physics. I don't know Salas, but you don't NEED Apostol to get started. You need to get a feel for calculus that is usually more difficult to come by when studying the rigorous version. Newton didn't talk about limits and continuity the way that we do today.

At some point, if you are really interested in pursuing physics, you will need some more rigor, but IMO if you are discouraged studying Apostol, you will be much better served starting with a text that has more graphs and spells some things out more clearly.

3. Mar 13, 2015

### shinobi20

I have already taken Calculus in a "Stewart" style, that is why I am re-learning calculus from a rigorous way, but my question is HOW rigorous should I be knowing I want to pursue theoretical physics (probably in the field of quantum field theory). I know that Apostol is for the pure mathe majors but I'm confused why a lot of posters here recommend apostol for physics majors.

4. Mar 13, 2015

### DrewD

Oh, okay. Then someone with more knowledge of Salas should probably help you. I don't know what is covered in that text.

5. Mar 18, 2015

### verty

I would argue, it's not how rigorous but rather how thoroughly or how quickly you should learn it. Being able to answer any question you could see on an exam is one measure of knowledge in a subject, but that is not enough. What you need is to be able to answer any question at some time in the future, and being able to answer those questions now is not a guarantee that you'll be able to answer them in the future.

That is the difference between the "Stewart" style recipe-book learning that one is likely to forget, and a more rigorous understanding that is likely to stay there forever. So your goal should be to gain a knowledge of calculus that is permanent because it all fits together and makes sense, and because it is constructed from simple parts and is justifiable, in the sense that one can justify why something is like it is. Why does integration by parts work, etc.

So that is my answer to how rigorous it should be: rigorous enough that you won't forget it. That said, rigorous books can be difficult to learn from so part of the challenge is to learn how best to learn from such books.

And I mentioned that how quickly you should learn it is an important question. I'm sure theoretical physics has a ton of stuff you would have to learn and in a hurry. So speed is paramount. My question is this, how quickly can you reach a level of knowledge that is permanent and makes calculus a tool that you can wield when needed?

To be more specific, I would aim to be as quick as possible, take as many shortcuts as possible, and get yourself to the level needed, being able to answer any calculus question that you would find in a typical book like Stewart and understanding what calculus is about. And if you have questions like what $dy \over dx$ really means, find the answers.

And once you get there, move on to the multivariable stuff and race through it as well. Try to keep in mind that you want to learn physics, not calculus. Calculus is going to be a tool and using it to answer questions is the point.

6. Mar 18, 2015

### shinobi20

Thanks for that wonderful insight but a lot of posters here are saying that when someone is going to do research on theoretical physics, say, QFT, studying from apostol/spivak is a must in contrast to stewart style books. I'm confused, I agree with your advice that I should learn calculus fast because there are so much topics in physics to cover and it seems unlikely for someone to master the rigor of calculus given the vast number of physics one needs to learn. Because I'm thinking that is it not possible to learn for example group theory WELL given someone learned calculus from stewart or salas.

7. Mar 18, 2015

### DrewD

If you want to do research in QFT, you will need much more beyond Apostol or Spivak. I don't do research in QFT, but I'm pretty sure that you will need to spend a lot of time doing much more complicated math than you will find in Spivak. The point that verty makes that is important is that "rigorous" (one of my profs used to ask if anyone knew a rigorous definition for rigorous and if the term "well-defined" was well defined) books can be difficult to get through and slow going. You don't need rigorous math to get through quite a lot of physics, but eventually you will. If you want to study rigorous math now, do that. If you are not yet finding it a hindrance, wait. Even though I don't know the specifics of the Salas book, I think this is reasonable advice.

As for group theory, you don't need any calculus. Often examples from basic calculus are included, but they could be skipped. Group theory is very simple... until it suddenly isn't.

Edit: I realized that maybe you think Stewart won't prepare you for proofs in a group theory book. It won't, but who cares. Learn to write proofs and group theory at the same time!

8. Mar 18, 2015

### shinobi20

Thanks verty and DrewD for the wonderful advice. So I will stick with Apostol all the way.

9. Mar 29, 2015

### laramman2

I have used Salas, Hille, Etgen for my elementary calculus classes. I don't think it is enough. It doesnt even cover all of the important concepts and techniques in differential equations (btw, it only covers ode).