Why Study Analysis As A Physics Student

I started learning differential geometry without a sound base on calculus. Here's what happened afterwards.

I was solving a problem on vector calculus from Lovett's Differential Geometry of Curves and Surfaces. It involved solving a first order ODE but at first there was a square of the first derivative. All I had to do is take the square root of both sides and solve a very easy first order ODE.

But, what I did was both funny and nonsense. I treated the square of the first derivative as a 2nd derivative and started solving a 2nd order ODE.

To add more to the irony I sent the solution to a graduate student for feedback. Thus, the quote on the picture of the heading was born.

And then he sent me a long reply on why should I study calculus with great rigor and details for a long time. I became agitated as I took it as an insult. I didn't think there was anything left for me to learn in "elementary" calculus and/or analysis. All the high school level calculus I knew seemed to me was enough for to do physics.

If only I could be less wrong.

As if it was not enough then this incident happened. I tried to solve a improper integral involving arc tangent function. I made a hilarious mistake on this problem too. The way I solved it needed to deal with 2 separate divergent integrals. But though adding two divergent quantities together necessarily gives another divergent quantity; in general, subtracting one from the other doesn't always give off zero.

Infinity minus infinity isn't always zero 

So, I have started learning analysis thoroughly now.

But, these problems are some mathematically challenging problems, right?  Does it really matter for a physics student to learn analysis with full rigor rather than just learning some integration and differentiation techniques?

It solely depends on what kind of physics anyone wants to do. If anyone wants to be an experimental physicist, he might skip all those nitty-gritty details of analysis. But, for theoretical/mathematical physicist, perhaps mathematician Terence Tao on this has put it beautifully on his Analysis I book:

It is a fair question to ask, “why bother?”, when it comes to analysis. There is a certain philosophical satisfaction in knowing why things work, but a pragmatic person may argue that one only needs to know how things work to do real-life problems. The calculus training you receive in introductory classes is certainly adequate for you to begin solving many problems in physics, chemistry, biology, economics, computer science, finance, engineering, or whatever else you end up doing - and you can certainly use things like the chain rule, L’Hˆopital’s rule, or integration by parts without knowing why these rules work, or whether there are any exceptions to these rules. 

However, one can get into trouble if one applies rules without knowing where they came from and what the limits of their applicability are. Let me give some examples in which several of these familiar rules, if applied blindly without knowledge of the underlying analysis, can lead to disaster.

The first chapter of Tao's book Analysis I has some very nice example of what can happen if you use analytical tools carelessly. And you have probably  have understood what blunders one can make by reading about the examples I have set.

Moreover, it's not just about doing your algebraic manipulations right. If one needs to develop/improve a theory, one has to first learn all the foundation of the prior theories. Without the mathematical rigor of analysis which has been a centerpiece of all the theories of physics to this date, developing or improving a theory isn't possible. As Gerard t' Hooft has put it in his Theoretical Physics as A Challenge :

Theoretical Physics is like a sky scraper. It has solid foundations in elementary mathematics and notions of classical (pre-20th century) physics. Don't think that pre-20th century physics is "irrelevant" since now we have so much more. In those days, the solid foundations were laid of the knowledge that we enjoy now. Don't try to construct your sky scraper without first reconstructing these foundations yourself. The first few floors of our skyscraper consist of advanced mathematical formalisms that turn the Classical Physics theories into beauties of their own. They are needed if you want to go higher than that.

Perhaps, now it has become clear why you should study analysis with style. All that being said here are some nice books about on calculus + analysis :

1. Calculus - Howard Anton
2. Calculus - James Stewert
3. Calculus I & II - Apostol
4. Analysis I - Terence Tao
5. Analysis II - Terrence Tao
6. Mathematical Analysis II - Vladimir A. Zorich (Not a beginner text)

All these incidents made me learn one thing. If you are in a hurry, physics may not be the thing for you.

To finish, as a theorist wrote on a post at physics.stackexchange.com:

Theoreticians, in my opinion, should study mathematics like math majors, almost forgetting about physics for a time; this is the point I feel so strongly about. The thing is that math is such a big subject, and once you have the road map of what is important for theoretical physics; then it really takes years of study to learn all the mathematics. I think it's so bad how many physics professors, who are themselves experimentalists, teach math improperly to young theoretician's. I personally had to unlearn many of the things I thought I knew about math, once I took a course based on Rudin's "Principle's of Analysis".