Is analysis necessary to know topology and differential geometry?

[-Dragoon-]

I'm a physics major interested in taking some upper level math classes such as topology, differential geometry, and group theory but these classes are only taught in the math department and are heavy on the proofs. Analysis are recommended and preferred prerequisites but are apparently not necessary. Should physics majors need to know analysis, especially before learning topology and differently geometry? If so, are there any good books you'd recommend on analysis for non-pure math majors and getting introduced to proofs?

Thanks in advance.

 

[TheKracken]

Do you have a heavy background in proofs? Topology is very heavy on proofs, but does not necessarily rely on the information you would learn in Analysis. If you are comfortable with proofs and the course does not require you to have analysis then go for it.

 

[WannabeNewton]

A lot of topology will lack motivation if you have never studied real analysis. For example, the definition of a continuous function $f$ between two topological spaces $X$ and $Y$ is: $f$ is continuous if for all open sets $U\subseteq Y$, $f^{-1}(U)$ is open in $X$. You may very well be asking yourself "how the hell did they even come up with this definition?!". Of course if you've studied basic real analysis you would know that the $\epsilon$-$\delta$ definition of continuity is trivially equivalent to the above definition for metric spaces but the above definition makes no reference to metrics and hence can be lifted to topological spaces. This is a very simple example but the motivation for a lot of concepts in topology comes from real analysis.

That being said, you don't technically need real analysis to study topology. I studied topology before I did anything more than continuity and convergence in the real analysis setting and I was fine (but to be fair I had a really awesome teacher). I should also add that the way proofs are done in topology is quite different from the way they are done in real analysis if one is to stick to epsilonics/metrics.

The same goes for differential geometry although to a much lesser extent. Topology is an absolute necessity for differential geometry though (meaning the most general form of differential geometry and not differential geometry of curves and surfaces).

Regardless, in my opinion real analysis is much, much, much more fun than differential geometry (but not as fun as topology!) so take from this what you will.

The reason many departments keep real analysis as a pre-req is that real analysis is the first proper proof heavy course that students tend to take and hence serves as a stepping stone into more advanced proof heavy courses (topology, functional analysis, measure theory etc.). If you've had a lot of experience with proofs then I would echo what TheKracken said. Otherwise I would advise against jumping straight into a topology course.

If you fall into the latter category then there are a lot of great real analysis books out there for you to choose from. My personal favorite is "Real Analysis"-Carothers as it is basically an ~400 page problem solving book. There are also the classics by Apostol and Rudin. Another good choice is "Introduction to Analysis"-Rosenlicht.

 

[espen180]

Differential geometry is locally (multivariable) real analysis, so it is absolutely necessary. For example, many basic results use the inverse and implicit function theorems, and the very definition of a manifold assumes you know basic multivariable real analysis. In addition, the whole point of an introductory course in differential geometry is to lift the machinery of regular real analysis to the manifold level.
So you should definitely take a real analysis course and a topology course (if possible, in parallell) before diving into differential geometry.

 

[WannabeNewton]

espen's post reminded me to point out that you also need to know a very good amount of theoretical linear algebra before delving into the theory of differentiable manifolds.

 

[-Dragoon-]

Thanks for all the advice. I was planning on getting Rosenlicht introduction to analysis as I like the dover books a lot and they're also well within my budget. Most of the other books that have been recommended are stupendously expensive relatively speaking, but would it be a sufficient introduction to analysis to start tackling topology and differential geometry? I was also looking into getting Velleman's How to Prove It just to get the hang of reading and writing proofs.

 

[-Dragoon-]

espen's post reminded me to point out that you also need to know a very good amount of theoretical linear algebra before delving into the theory of differentiable manifolds.

Which books would you recommend, then? I know Axler's "Linear algebra done right" is considered the standard, but it's unfortunately not within my means to buy it at this time.

 

[R136a1]

Which books would you recommend, then? I know Axler's "Linear algebra done right" is considered the standard, but it's unfortunately not within my means to buy it at this time.

You can always use this free book: http://www.math.brown.edu/~treil/papers/LADW/LADW.html
Another good choice is Lang's linear algebra. Personally, I'm not a fan of Axler since he shuns determinants so much.

That said, you can do differential geometry without analysis. A very good book here is: https://www.amazon.com/dp/184882890X/?tag=pfamazon01-20 It just requires some calc 3 and some basic linear algebra. Obviously, you won't be going very deep and you won't do manifolds and bundles. But you can get the taste of some basic differential geometry, and you will find advanced texts much easier.

For analysis, I recommend Rosenlicht or Berberian: http://books.google.be/books/about/A_First_Course_in_Real_Analysis.html?id=pvI1DFVgP9UC&redir_esc=y You won't need much more than this.

Of course, to study the advanced texts, you're going to need some topology. Lee's topological manifolds book is excellent. And you can follow-up by his excelent smooth manifolds book.

 

[Astrum]

You can always use this free book: http://www.math.brown.edu/~treil/papers/LADW/LADW.html
Another good choice is Lang's linear algebra. Personally, I'm not a fan of Axler since he shuns determinants so much.

I can understand this view point, but I prefer Axler over a text such as Hoffman and Kunze. It (Hoffman) seemed too wordy, where Axler is generally right to the point. Hoffman and Kunze is a good text, and if you're worried about determinants, you should use it as a secondary source. Never checked out Lang, so I can't really offer anything on that.

 

[espen180]

I second Lee's topological manifolds as a good book for topology. Follow up with his smooth manifolds book and you have a solig grounding in differential topology and you can dive head-first into differential geometry.

For linear algebra, Steven Roman's book "Advanced linear algebra" is the gold standard, but it may be too advanced for you if you haven't seen any abstract algebra before.

@R136a1: Looks like Presseley's book gets mixed reviews on amazon. Are you sure the classic https://www.amazon.com/dp/0486667219/?tag=pfamazon01-20 isn't a better choice?

 

[R136a1]

I second Lee's topological manifolds as a good book for topology. Follow up with his smooth manifolds book and you have a solig grounding in differential topology and you can dive head-first into differential geometry.

For linear algebra, Steven Roman's book "Advanced linear algebra" is the gold standard, but it may be too advanced for you if you haven't seen any abstract algebra before.

@R136a1: Looks like Presseley's book gets mixed reviews on amazon. Are you sure the classic https://www.amazon.com/dp/0486667219/?tag=pfamazon01-20 isn't a better choice?

Whatever book you choose, don't get Kreyszig. The book is horribly outdated. I'm sure the book was good in the 18th century, but now things should be done very differently. Kreyszig's functional analysis text is brilliant, but his differential geometry text is now too outdated and the notation is old-fashioned (and trust me, notation matters a lot in differential geometry!).

Good books on elementary differential geometry are:

https://www.amazon.com/dp/0132125897/?tag=pfamazon01-20 A bit of a leisurely introduction to the topic. Very nice, and great exercises (with hints which tend to solve the entire problem).

https://www.amazon.com/dp/0120887355/?tag=pfamazon01-20 Very beautiful book which does things with forms. It does require some more maturity than Do Carmo and Pressley.

https://www.amazon.com/dp/0132641437/?tag=pfamazon01-20 Is nice too.

 

[-Dragoon-]

You can always use this free book: http://www.math.brown.edu/~treil/papers/LADW/LADW.html
Another good choice is Lang's linear algebra. Personally, I'm not a fan of Axler since he shuns determinants so much.

That said, you can do differential geometry without analysis. A very good book here is: https://www.amazon.com/dp/184882890X/?tag=pfamazon01-20 It just requires some calc 3 and some basic linear algebra. Obviously, you won't be going very deep and you won't do manifolds and bundles. But you can get the taste of some basic differential geometry, and you will find advanced texts much easier.

For analysis, I recommend Rosenlicht or Berberian: http://books.google.be/books/about/A_First_Course_in_Real_Analysis.html?id=pvI1DFVgP9UC&redir_esc=y You won't need much more than this.

Of course, to study the advanced texts, you're going to need some topology. Lee's topological manifolds book is excellent. And you can follow-up by his excelent smooth manifolds book.

I recently bought the analysis book by Rosenlich, it's at a very good price for a book that appears to be really good, which is hard to find in a math or physics book these days.

However, it's quite compact at only 248 pages. Is this really all the real analysis I'd need to take on higher texts in topology and differential geometry? Does this include group theory as well?

Also, what about any advanced linear algebra (more advanced and rigorous than physics' majors usually cover such as in Axler's book)? Or is basic linear algebra good enough?

 

[R136a1]

I recently bought the analysis book by Rosenlich, it's at a very good price for a book that appears to be really good, which is hard to find in a math or physics book these days.

However, it's quite compact at only 248 pages. Is this really all the real analysis I'd need to take on higher texts in topology and differential geometry? Does this include group theory as well?

Yes, it's really everything you need, although it might be helpful to read up a bit on metric spaces, I don't think Rosenlicht covers that.
It doesn't do group theory, but there's not really much group theory you need anyway. The thing is that many group theory books start of explaining finite groups, while the groups in differential geometry are usually infinite. So group theory texts aren't really all that useful.

Also, what about any advanced linear algebra (more advanced and rigorous than physics' majors usually cover such as in Axler's book)? Or is basic linear algebra good enough?

If you know things like Axler, then you're all set. You don't need more than that. You certainly do need to be acquainted with vector spaces and linear transformations though.

 

[-Dragoon-]

Yes, it's really everything you need, although it might be helpful to read up a bit on metric spaces, I don't think Rosenlicht covers that.
It doesn't do group theory, but there's not really much group theory you need anyway. The thing is that many group theory books start of explaining finite groups, while the groups in differential geometry are usually infinite. So group theory texts aren't really all that useful.
If you know things like Axler, then you're all set. You don't need more than that. You certainly do need to be acquainted with vector spaces and linear transformations though.

Thanks. Just one last question, I've been looking at that differential geometry book you recommended that only requires calc III and basic linear algebra. Is there an equivalent book for topology that you'd recommend?

Edit: The Rosenlicht book actually does cover metric spaces.

 

[TomServo]

Not OP but do I need to go to all of this trouble to do GR? Will a grad-level GR class cover the necessary math or will I need to study topology first?

 

[WannabeNewton]

Not OP but do I need to go to all of this trouble to do GR? Will a grad-level GR class cover the necessary math or will I need to study topology first?

You will not have to go to all this trouble. Tensor calculus and tensor algebra are done very differently in most GR texts so most of what you learn from the aforementioned texts won't even be of much help to you in the end as far as solving GR problems goes.

 

[espen180]

Not OP but do I need to go to all of this trouble to do GR? Will a grad-level GR class cover the necessary math or will I need to study topology first?

From my experience, a first grad-level GR course will only deal with local phenomena, so you can pretend to be working on a topologically trivial manifold.

I suppose if you want to work in more exotic contexts, like on a non-orientable manifold, or in general any case where the problem of patching together local tensor fields into a global ones is non-trivial, you would need to bring more sophisticated tools to the table.

 

[-Dragoon-]

You will not have to go to all this trouble. Tensor calculus and tensor algebra are done very differently in most GR texts so most of what you learn from the aforementioned texts won't even be of much help to you in the end as far as solving GR problems goes.

Completely contrary to what 2 of my physics professors told me, and they're well respected cosmologists/relativists in the field. They have told me the best investment I can do as an undergrad in learning GR is taking topology and differential geometry, and that would put me well ahead of the competition when it comes to graduate school.

 

[WannabeNewton]

Go ahead and learn topology and differential geometry from the aforementioned math texts, open up a GR text like Wald, and see how much of what you learned actually has any relevance in solving the end of chapter problems. I can tell you from experience that not even the first 4 chapters of Lee's text on topological manifolds (which I went through thoroughly) had much if any relevance except for some basics that showed up in chapter 8 of Wald's text (global causal structure). I don't think you realize how different the math in a math textbook is from what is presented in a physics textbook. You have to take a look yourself to see.

Even better, after you spend all your time on Lee and/or Rosenlicht go ahead and attempt the problems in MTW (which are far superior to the problems in Wald) and see just how inconsequential what you learned from the aforementioned math texts actually is in solving the problems.

As a side note, I like to learn pure math for the sake of pure math. Thinking that learning pure math will somehow help you understand a physics text better than someone who doesn't have a background in pure math is absolutely ridiculous.

EDIT: In fact what going through various pure math texts has really done for me is make it really hard to go through physics textbooks without being nitpicky about every single detail that the physics textbooks get wrong from a mathematical standpoint. Go through a functional analysis text like Conway and try to read a standard QM text (like Sakurai or Shankar) and take note of how impossible it is not to burn the text because of how badly it butchers the math.

 

[TomServo]

I hope that one day, I too will annoy mathematicians. I think I'll use the approximation e=pi since they're close enough. Maybe I'll say L'Hospital a LOT.

 

[espen180]

@WannabeNewton: I hope you're not arguing that learning the proper mathematics is useless because it isn't necessary in the end-of-chapter problems in your textbooks! When the time comes to actually do some research, I'm sure you'll be glad you studied it.

 

[R136a1]

@WannabeNewton: I hope you're not arguing that learning the proper mathematics is useless because it isn't necessary in the end-of-chapter problems in your textbooks! When the time comes to actually do some research, I'm sure you'll be glad you studied it.

Depends on the research. There is enough research in physics where pure mathematics is useless.

 

[espen180]

Depends on the research. There is enough research in physics where pure mathematics is useless.

Of course, but does that include theoretical general relativity?

 

[WannabeNewton]

When the time comes to actually do some research, I'm sure you'll be glad you studied it.

How so? Mathematical physics is different from theoretical physics (and the term "theoretical" is not even connoted properly in informal discussions). Can you give examples of theoretical GR research wherein a thorough study of e.g. Spivak's opus on differential geometry comes into play over a standard graduate GR text?

 

[espen180]

How so? Mathematical physics is different from theoretical physics (and the term "theoretical" is not even connoted properly in informal discussions). Can you give examples of theoretical GR research wherein a thorough study of e.g. Spivak's opus on differential geometry comes into play over a standard graduate GR text?

Although I'm not aquainted with the full content of Spivak, the following popped up after a quick search. You can tell me whether the material is contained within standard textbooks or not.
http://arxiv.org/abs/0809.3596
http://arxiv.org/abs/gr-qc/0402105
http://arxiv.org/abs/hep-th/9706092

For the record, the version of tensor calculus taught in most GR classrooms is the proper tensor calculus, it is just the notation which is different. However, surely you do realize that many advances in areas like differential geometry are caused by a need for them in physics (Witten comes to mind as a good source of ideas), so of course you need to know the necessary mathematics if you want to have any hope of advancing the field, and not just study particular models in it.

 

[WannabeNewton]

All of that can be learned just from Nakahara's text. Again, I never said that a formal study of pure math is useless to a physics student; I said that claiming a formal study of pure math to be necessary for the sake of physics is fallacious. Seriously, if you were to spend a semester studying e.g. Bredon's differential topology text with the mindset that all of what you learn in that text will actually be relevant to physics then you have another thing coming. This is why texts like Nakahara, Baez, and Frankel exist.

 

[espen180]

But isn't Nakahara basically a math book but without the proofs? I think it is a stretch to call is a physics book, even if it is standard in the physics curriculum.

I said that claiming a formal study of pure math to be necessary for the sake of physics is fallacious.

And I agree, insofar as "pure math" refers to the business to proving theorems.

 

[-Dragoon-]

Go ahead and learn topology and differential geometry from the aforementioned math texts, open up a GR text like Wald, and see how much of what you learned actually has any relevance in solving the end of chapter problems. I can tell you from experience that not even the first 4 chapters of Lee's text on topological manifolds (which I went through thoroughly) had much if any relevance except for some basics that showed up in chapter 8 of Wald's text (global causal structure). I don't think you realize how different the math in a math textbook is from what is presented in a physics textbook. You have to take a look yourself to see.

So you're basing the utility of the math knowledge and intuition on solving back of the book problems? Are you really being serious here? Using your logic, why should physics majors take a course in partial differential equations if it won't directly help them solve the problems dealing with Laplace's Equation out of Griffith's?

I don't know about you, but for me personally math intuition and physics intuition goes hand in hand. I truly never feel I have a deep understanding of a physics concept unless I have an even deeper understanding of the math behind it. Also, have you done any research in the field? I'm pretty sure GR specialists who strongly suggested I take these classes know much more than you do when it comes to the field, with all due respect of course.

 

[WannabeNewton]

But isn't Nakahara basically a math book but without the proofs? I think it is a stretch to call is a physics book, even if it is standard in the physics curriculum.

It isn't a physics book definitely and it's arguably a math book in the sense that it discusses a slew of connected math topics. But it isn't like the other books mentioned previously in the thread in terms of its purpose and utility to a physics student (more or less relating to what you said directly below).

And I agree, insofar as "pure math" refers to the business to proving theorems.

Yeah exactly what I meant by "a formal study of pure math" since all the analysis and topology texts so mentioned in this thread are of this nature. They can definitely be sufficient but they aren't necessary and someone who studied Bredon's differential topology text wouldn't be able to claim superiority over someone who stuck to Nakahara when it comes to physics (not including mathematical physics of course because that's a whole different ballgame).

 

[WannabeNewton]

So you're basing the utility of the math knowledge and intuition on solving back of the book problems? Are you really being serious here? Using your logic, why should physics majors take a course in partial differential equations if it won't directly help them solve the problems dealing with Laplace's Equation out of Griffith's?

Hah. I clearly said "a formal study of pure math". You haven't seen a true pure math course in PDEs before have you? It's nothing like the kind engineers and physics students generally take. You haven't done nearly enough to know the difference between math as used in physics and math as used in a pure math text as this thread clearly shows. There's nothing wrong in learning what you want to learn but don't go around making it seem like studying Papa Rudin, Lee's topological manifolds, Rosenlicht etc. will make your grad school endeavors somehow easier than someone who just stuck to mathematical methods books and the likes constructed primarily for utility in the line of physics.

 

[jesse73]

The physics books that cover mathematical topics also tend to add more physics insight to the problem. A book on mathematical methods for physics will make sure you know why eigenvalues are important in QM while a book focused on just mathematical concerns won't even mention QM.

There is a difference in a linear algebra or math class for mathematicians course and a linear algebra class meant for engineers or physicist. There is a even difference in a QFT class taught by particle theorist and one taught by condensed matter theorist. You should try to take the one more directly related to your goals so you get a higher rate of return for your time invested.

 

[R136a1]

I'm pretty sure GR specialists who strongly suggested I take these classes know much more than you do when it comes to the field, with all due respect of course.

WannabeNewton is one of our best posters when it comes to GR. I wouldn't immediately dismiss his advice like this.

https://www.youtube.com/watch?v=obCjODeoLVw

And a good knowledge of pure math can sometimes completely destroy your love and ability to do physics. I have significant difficulties in solving physics problems because I think way too mathematical about them.

 

[Student100]

So you're basing the utility of the math knowledge and intuition on solving back of the book problems?

No, I believe he's basing his argument on the utility of pure math in garnering any kind of advantage in physics over someone who just studies the utility of math.

But maybe my reading comprehension is folly.

 

[-Dragoon-]

Hah. I clearly said "a formal study of pure math". You haven't seen a true pure math course in PDEs before have you? It's nothing like the kind engineers and physics students generally take. You haven't done nearly enough to know the difference between math as used in physics and math as used in a pure math text as this thread clearly shows. There's nothing wrong in learning what you want to learn but don't go around making it seem like studying Papa Rudin, Lee's topological manifolds, Rosenlicht etc. will make your grad school endeavors somehow easier than someone who just stuck to mathematical methods books and the likes constructed primarily for utility in the line of physics.

The main problem I have is actually finding books geared towards physicists that go beyond very basic and trivial stuff like Fourier transforms, contour integration, Cauchy's theorem, special functions, and basic PDE methods. I've taken the math methods for physicists course in the department, and despite getting an A+, I still feel I don't know any "serious" mathematics.

The only book I've found thus far that does exactly what you said is "geometrical methods of mathematical physics" by Schutz, but it really only does serious treatment differential geometry. Before graduate school, I at least want to have a very deep understanding of complex analysis, basic group theory, topology, and differential geometry. I enjoy doing math for the sake of math and it's mostly curiosity, but you mean to absolutely tell me that having a deeper understanding of these topics than your typical graduate student won't be a boon at all? So my professors simply have no idea what they are talking about?

 

[R136a1]

The main problem I have is actually finding books geared towards physicists that go beyond very basic and trivial stuff like Fourier transforms, contour integration, Cauchy's theorem, special functions, and basic PDE methods. I've taken the math methods for physicists course in the department, and despite getting an A+, I still feel I don't know any "serious" mathematics.

The only book I've found thus far that does exactly what you said is "geometrical methods of mathematical physics" by Schutz, but it really only does serious treatment differential geometry. Before graduate school, I at least want to have a very deep understanding of complex analysis, basic group theory, topology, and differential geometry. I enjoy doing math for the sake of math and it's mostly curiosity, but you mean to absolutely tell me that having a deeper understanding of these topics than your typical graduate student won't be a boon at all? So my professors simply have no idea what they are talking about?

Group theory: mathematics courses usually focus on finite groups, while physicists usually use infinite matrix groups. For this reason, math courses on group theory do not tend to be very useful.

Topology: physicists tend to focus on very nice spaces such as manifolds. A topology course deals with very weird and strange spaces, and many of it is not very useful at all outside of mathematics. A deeper understanding of compactness and connectedness is useful though.

Differential geometry: this could be useful to physicists as some kind of foundational thing. But still, I fail to see how a rigorous proof of Whitney's embedding theorem could benefit you. Plus a differential geometry text in the pure math setting will strive to stick to index-free, coordinate-free methods throughout and this will not fly in GR.

Complex analysis: I don't see why physicists need to bother with the many proofs in this course. Sure, it is a very elegant course and the proofs are beautiful. But I doubt it will make you a better physicist. Physicists who spend most of their time worrying about whether interchanging series and integral is allowed don't produce much physics.