# How does calculus tie in for physics with other math (diff geometry, linear alg)?

1. Aug 4, 2004

### singleton

I will be entering undergraduate studies in 2 years time for physics.

Between now and then I would like to teach myself the mathematics so I have a solid foundation entering university and get ahead.

Right now I'm finishing up OAC/Grade 13 calculus (via correspondence...teaching myself from books and I go in to write the tests/submit my papers) so my knowledge of calculus is limited.

Along with calculus, the undergraduate programs include linear algebra and I've heard others recommend differential geometry (by the way, are there other maths I should learn?). Being limited in math I don't know how these all tie in together. Can I go straight through and focus on one at a time--calculus all the way up to II/III, then do linear algebra and then differential geometry?

Calculus is the center for the physics program (it has 3 classes whereas the others usually are one) so I figure I should post it here because you'd know how it ties together ;)

Thanks!

2. Aug 4, 2004

### Gza

You seem to have the right idea. Just stick with getting really good at calculus through your lower division years, since you'll be pretty much expected to have a solid grasp of it (single thru multvariable) when entering higher math classes.

3. Aug 5, 2004

### theFuture

as Gza said, calculus should be your focus. You should first plug all the way through that. Courses in linear algebra and differential equations should be next on your list. Beyond that, it really depends what you're specializing in to determine what sort of math you should take next. You can do diff. eq. after 1 or 2 semesters of of calculus and LA ties in nicely with diff. eq.

4. Aug 5, 2004

### Feynman

If you like continue your studies on Theoretical physics, you must learn : topology,
algebric topology,Partial differential equations,algebric geometry,and differential geometry
Calculus is the first stap but it isn't enouph

5. Aug 5, 2004

### theFuture

Feynman: Those are good recommendations but I'm not sure you need to be taking a whole lot of math classes to become a good physicist. I believe all the math you need the physics department will teach you (or at least that's been my experience).

6. Aug 5, 2004

### singleton

Thanks guys I appreciate the feedback.

My plan is over the next two years to learn not only the math that I will take in undergrad studies but also to go further with it and learn more.

A personal goal I'd like to understand astrophysics, general relativity and big picture things but I'd really like to have a good basic understanding in each field to be well rounded. 75% master of one and 25% jack of all trades I guess you can say ;)

But thanks again for the tips! I'm planning to pick up Calculus Made Easy and the How to Ace Calculus books next. I don't expect to learn much more from them than I already know (from OAC calculus) but hopefully, through repetition and alternative teachings, I can better my calculus understanding at this level. :)

cheers

7. Aug 6, 2004

### Feynman

TheFutur I'm talking of PHD studies on theoretical physics
Singleton if you need study general relativity , differential geometry is essential
FEYNMAN

8. Aug 6, 2004

### singleton

Oh okay. That and advanced calculus knowledge eh? When should I begin that, like how far into studying calculus? And what about linear algebra *hate to get off topic* is that necessary?

9. Aug 6, 2004

### mathwonk

I am a mathematician. if you want to be a physicist I recommend you go to the local college and talk to some physics professors. Or maybe read the Feynman lectures on physics. But why listen to us? You do not even know who we are?

10. Aug 7, 2004

### singleton

I recognize this is a math forum but my question had more to do with math then physics in that I wanted to know where the maths tie in (as in, together with each other, not with respect to physics) and the order in which to learn them.

Why listen to you? I don't know who you are, that is right. But do you ever know someone? ;)

11. Aug 7, 2004

### mathwonk

Dear Mr Singleton and Mr. Feynman,

Let me apologize for implying that not only I, but also others here may not be highly qualified in every area we comment on. I only know that to be true of myself, but I confess that in my arrogance, especially late at night and after wine, I have implied questions as to some other people's reliability as well. Mr Feynman is quite right to challenge me on this.

I confess too I do not even know what a second master of theoretical physics is, but it is clear Mr. Feynman has a great deal of personal experience and expertise both in theoretical physics and in studying for it. Thus it is no doubt accurate that any one wanting to follow his footsteps is well advised to learn differential geometry at some point.

To Mr. Feynman, in spite of my rudeness in the way I suggested it originally, I wonder if perhaps you/we could be of even more service to Mr Singleton by proposing something like a timeline for studying these topics. My worry was that we might discourage someone who is now at the Calculus Made Easy stage, by loading on too much too soon.

Of course I admit that a teacher who really understands it can probably make differential manifolds seem natural even to a beginning calculus student.

As to my own specialty I like geometry and calculus, and algebraic geometry, and complex analysis of one and several variables, and algebraic topology, but am far from a master of any of them. I am thoroughly ignorant of physics, but very impressed by the wonderful intuition physicists seem able to bring to bear even on apparently purely mathematical questions, such as the recent breakthroughs in enumerative algebraic geometry wrought by the theory of quantum gravity.

Please feel free to modify for Mr Singleton's benefit the following elementary remarks which are solely from my own perspective:

As to how the various topics tie in,
1) (differential) calculus is usually now regarded as a method of approximating non linear phenomena by linear phenomena, the simplest example being the approximation of a curve, (the graph of a function), near a given point, by its tangent line (the graph of its derivative).
In two variables one approximates a surface, (the graph of a function of two variables), near a given point, by a tangent plane (again the plane is the graph of the derivative, which is a linear function of two variables).

In higher dimensions, not being able to picture the situation so easily, we again approximate non linear phenomena, by things we call "linear" phenomena.
This time we use an algebraic definition of "linearity" to free us from the need to visualize it. I.e. a function L is linear if it preserves addition and scaling, which means that L(x+y) = L(x) + L(y), and L(cx) = cL(x). This definition, and an appropriate definition of "approximates" allows us to do calculus, i.e. linear approximation, in arbitrary dimensions, and even in infinite dimensions.

2) Obviously, since calculus is the science of approximating non linear phenomena by linear ones, it is essential to understand first linear phenomena. This is why you must study linear algebra, essentially as a precursor to higher calculus. In low dimensions calculus is taught without pointing out this connection, because the linear phenomena are so simple as not to need special study, or because some of us were born before the linear algebra revolution, and simply teach the way we learned.

Clearly one cannot even define linearity unless the space on which the functions are defined has a linear structure, i.e. unless you can add x+y in it, so linear algebra is necessarily carried out on linear or flat spaces.

3) Unfortunately, or interestingly as it may be, space time is not flat according to Einstein who postulated that mass causes it to be curved (here I am on thin ice, and out of my depth, so please help me out here Feynman). Thus how can we use calculus to study general relativity? or space time at all?

The answer is to note that it is not the original non linear function that must be defined on a linear space but only its derivative, i.e. its linear approximation. So here come the idea of differential manifolds, a differential manifold is a curved space, like a curved surface on which one has defined non linear phenomena of interest, and then one proceeds in two stages to approximate it by linear phenomena.

First one approximates the curved space itself by a flat space, which is again called the tangent space, and then one approximates the original non linear function which was defined on the curved space, by a linear object which is defined on the tangent space.

For instance if we regard the surface of the earth as a sphere, the sphere is the curved space, and we could be interested in the phenomenon of motion of water on the surface of the earth. To approximate this linearly, we fix a given point at which to make our study, say the north pole, and approximate the sphere there by a flat tangent plane. Better yet we put a (different) tangent plane at every point of the sphere.
Then we approximate the flow of water on the sphere by assigning at each point, a velocity vector in the direction of the flow, which vector lies in the tangent plane at the given point. This "vector field" is analogous to the derivative of a function in calculus.

4) The theory above is called differential manifolds, but the theory of differential geometry goes one step further and adds a notion of length of tangent vectors, usually called a "metric" or Riemannian metric on the tangent bundle as Mr Feynman correctly said. This enables one to compare the extent of curvature at different points of our space, hence to distinguish a large sphere from a small one, such as a small moon around our planet.

I am not too up on differential geometry, but the key concept seems to be curvature which is measured by an object called a "tensor" which comes from "multilinear algebra" (the second course in linear algebra). Basically the "first" derivative is a linear object and the "second" (or higher) derivatives are bilinear (or multilinear) objects.

(Basically, linear and multiplinear mathematics are easier than non linear mathematics, and so the main job in many areas of math is merely to approximate non linear objects by linear ones, or by families of linear ones, and so one simply must study as much linear maths as possible.)

5) Further, not only is space not flat, but it is not as simple as a sphere either, possibly taking more exotic shapes like the surface of a doughnut or a three dimensional analog of the surface of a doughnut, with many (possibly "black") holes of varying dimensions. Then one wants to study these holes, or "connectivity" phenomena as well. This subject is called algebraic or differential topology. I.e. the subject is topology, and one studies it by various tools available, algebra or calculus.

Now as to sources for studying these things, if you already know some calculus you will learn little or nothing of value, except some humor, by reading either Calculus Made Easy or Street Wise Calculus, (and I admit that I object on intellectual grounds to books such as the latter).

The first and still most rigorous and advanced introduction to calculus with linear algebra is the book Foundations of Modern Analysis by Dieudonne, from which sophomore (!) honors calculus used to be taught at Harvard in the 1960's. This book covers metric spaces, Hilbert space, calculus on Banach spaces, one complex variable, and some differential equations including Sturm Liouville theory, but nothing on manifolds. There are no figures in it, as Dieudonne was so strict a mental master as to disbelieve in allowing illustrations to learners. Fortunately his example is not much followed.

An easier source for learning is the book by Marsden and Tromba, written since then as an attempt to render this topic genuinely accessible to good college students, and used not too long ago at Berkeley. (Marsden was also a calculus instructor, and very well liked, at Harvard in the 60's). I have even taught from this book to high school students, one of whom is now a math prof (in topology) at a major university. It begins with a review of linear algebra, which Dieudonne assumes. It is also restricted to finite dimensions.

For a short succint introduction to "calculus on manifolds", it is hard to beat the book of that title by Michael Spivak. Spivak also has written the definitive textbook on differential geometry of our generation, in 5 volumes! available from his website at Publish or Perish. He is also a nice guy, brilliant, and a wonderful teacher. He himself was a differential topologist, with a construction in that subject named for him called the "Spivak normal fibre bundle", and he studied with both the greats John Milnor and Raoul Bott.

Algebraic topology can be hard to learn in my experience, but there is a fantastic book on differential topology by Guillemin and Pollack, a skillful rewrite for undergraduates of an even better but more advanced short book by John Milnor.
Spivak's differential geometry book also contains a wonderful section in volume one(?) on deRham cohomology, or algebraic topology taught from the perspective of calculus. Guillemin and Pollack also treat some deRham cohomology, which is basically path integration, largely borrowed from Spivak, at the end of their book, and they do a very nice job.

So may I suggest you read first Marsden and Tromba, then Guillemin and Pollack or Spivak's little book. Then maybe Spivak's first or second volume of differential geometry, and then take stock of your situation.

You might also try (for algebraic topology) the introduction by William Fulton, an algebraic geometer, who makes connections with other subjects in it nicely.

But by now someone else should chime in.

best regards,

roy

Last edited: Aug 7, 2004
12. Aug 8, 2004

### mathwonk

Mr Singleton,

I am interested to know, was my previous attempt to explain the connections between the various areas of calculus useful to you? If you like, I could add something on the connection between these topics and algebraic geometry and string theory, but my suggestion is that you probably already have enough on your plate for a while.

I was also hoping Mr. Feynman would contribute something to the discussion from his perspective as a physicist.

best regards

13. Aug 8, 2004

### singleton

Thank you very much! I think, for the most part I understood a lot of your post.

You recommended Foundations of Modern Analysis by Dieudonne, would Calculus volumes 1 and 2 by Tom Apostol be too advanced for me to use?

Thanks overall! There is much to learn so my focus for this year is to really work on calculus. I think I will focus on that all year and hopefully learn enough to be around a Calculus II or III stage given a full year during my off-college hours.

14. Aug 8, 2004

### mathwonk

Apostol's book on calculus, both volumes, is perhaps my absolute favorite for a top notch honors university calculus course. They use the second volume at Stanford but unfortunately do not teach the first volume, which makes life very tough on even their best entering students. You cannot do better than to go through as much of Apostol as possible.

Although the approach is high level, Apostol aims it at a (bright) beginner, so he goes out of his way to make it as motivated and understandable as possible. It is still tough going in places of course. You might also like to look at a copy of Courant, or Courant and John, or just go to the university library stacks and browse the calc books.

Actually I taught successfully from Apostol vol I, to a class of returning high school teachers once and they were very challenged and inspired by it. The thing about that book is, that even if you do not get through it perfecdtly the first time, you have still been exposed to the best version of the material, and can always return to it in future until it becomes clear. And eventually it will.

Good luck!

15. Aug 8, 2004

### mathwonk

apostol is very expensive new (over 100 dollars a volume) but there is a great used book site for math books:

http://www.abebooks.com/

I found two posts for copies of apostol there tonight: vol 1 at 30 dollars. and vol 2 at 18 dollars.

i recommend you try them and get your own copies as you will want to keep this book all your life.

16. Aug 8, 2004

### singleton

Thanks again everyone!

I think I will invest in a second hand set of Apostol books after I do do a few more practice books to solidify what I know.

They sound to be packed with difficult yet worthwhile content and hopefully with the interest I have I can get through them.

17. Aug 9, 2004

### Corneo

Mr. mathwonk

May I ask why is it not right to learn calculus from books such as "Calculus made easy" or "Street Wise Calculus"? I'll admit that these books lack the deepth and proof of what a top notch textbook offers.

18. Aug 9, 2004

### mathwonk

It is only my personal prejudice, and not entirely logical. I happen to like Calculus Made Easy very much, and have always recommended it, since encountering it as a young student. On the other hand, now that I am old, and have been subjected to a long sequence of books called "brain surgery for dummies" etc, I have grown cynical about their quality, and feel this type of title is now a marketing tool. In truth I am not very familiar with the contents of Street wise Calculus, but my impression is that it insults the reader's intelligence. I may be wrong. Some of my non mathematician friends like it.

As to Calc Made Easy, it was written by a fellow of the royal society of engineers, and is really very well done, at least in the earlier editions of 50-60 years ago. I do regret that Martin Gardner "modernized" and "improved" it some time back. But in general, one should learn from any source that helps him, her. In Mr Singleton's case, you recall he said he already knew some calculus, and these "made easy" books are intended for complete beginners.

Nonetheless, I still feel that the books written by masters are more helpful in the long run than those written by less qualified people. So for calculus, after getting past the fear of it in some way, I recommend the great books of our time, Apostol, Courant, Kitchen, Spivak. There is really no shortcut to learning a deep subject.

You will see a contradiction in my advice, however as every post I have made here is an attempt to render some deep subject in simpler terms than usually found in books, and I myself am not one of these masters, yet I give advice. So please consult any books you like, but you may be wise not to ignore the ones that most people say are best.

So in answer to your actual question "Why is it not RIGHT to learn from books like Streetwise...?" I would say rather it is probably not POSSIBLE to learn much from them, as they do not contain the real essentials of the subject, at least not for serious students.

In order to provide insight into a subject the author must himself have some. It is indeed difficult for most of us to match the insights of a master like Courant. In the old days in Europe, calculus books were written by the foremost mathematicians of the day, like Euler or Picard.

Still, learning is achieved by finding the right match between teacher and learner, and no one source is right for everyone, so there is always room for new books.

Unfortunately there is a tendency of calculus books in the US to be dumbed down in every subsequent edition, until they lose almost all value for good students. Thus the insights of current calculus book authors are often lost after the first edition. E.g. if I recommend Stewart to someone because I liked the 2nd edition, then the third edition comes along and is not as good as the second, because the publisher wants to sell even more books by making it even easier and less challenging. This also happened to Thomas and Finney between the 9th and 10 editions.

It is only the great books named above that are mostly immune to these harmful changes. Even Courant was completely rewritten as Courant and John at one point, and good as that book is, I prefer Courant in general.

Even the once excellent Schaums outline series on calculus has doubled in size over the years and yet become more shallow in content.

best wishes

Last edited: Aug 9, 2004
19. Aug 9, 2004

### Feynman

So Math,
you are right ,
Singleton have you an hotmail?
if yes give me your email to explain for you the necessairy Math for physucs
Feynman

20. Aug 9, 2004

### mathwonk

To Mr. Corneo,

Let me quote some of the introductory paragraph of Dieudonne's chapter VIII on differential calculus.

He says his presentation

"....aims at keeping as close as possible to the fundamental idea of calculus, namely the 'local' approximation of functions by LINEAR functions."

At this point one has already learnt something valuable from him. Indeed this has already answered one of Mr Singleton's principal questions about the role of linear algebra in calculus, and you will note that my answer to him was borrowed from this source where I originally learned it.

He goes on:

"In the classical teaching of calculus, this idea is immediately obscured by the accidental fact that on a one dimensional vector space, there is a one-to-one correspondence between linear forms and numbers, and therefore the derivative at a point is defined as a NUMBER instead of a LINEAR FORM."

"This...becomes much worse when dealing with functions of several variables: one thus arrives, for instance, at the classical formula giving the partial derivatives of a composite function, which has lost all trace of intutitive meaning, whereas the natural statement of the theorem is of course that the (total) derivative of a composite function is the composite of their derivatives, a very sensible formulation when one thinks in terms of linear approximations."

This beautiful summary of the meaning of the higher dimensional chain rule is unsurpassed in my experience. It already provides more insight than whole chapters written by lesser authors.

Here are a few useful words of advice from the introduction of Courant's book on a related matter:

"I should like to warn the reader specially against a danger which arises from the discontinuity [between school mathematics and university mathematics]. The point of view of school mathematics tempts one to linger over details and to lose one's grasp of general relationships and systematic methods. On the other hand, in the higher point of view there lurks the opposite danger of getting out of touch with concrete details, so that one is left helpless when faced with the simplest cases of individual difficulty, because in the world of general ideas, one has forgotten how to come to grips with the concrete. The reader must find his own way of meeting this dilemma. In this he can only succeed by repeatedly thinking out particular cases for himself and acquiring a firm grasp of the application of general principles in particular cases; herein lies the chief task of anyone who wishes to pursue the study of science."

I can add nothing of value to these words of a master teacher,

except: good luck to you!

21. Aug 9, 2004

### Corneo

Thank you for your response. I'll keep that in mind as I go through my math classes.

22. Aug 9, 2004

### singleton

Mathwonk I'm grateful for the feedback and insight you have given so far. It truly is priceless to me since I'm not in an institute right now and lack the guidance of a professor.

How long do you suppose someone could work through and have a solid understanding of the material in both Apostol volumes?

I'll be able to commit around 20-25 hours weekly into my schedule while I'm back at school this fall (computer programming diploma) for a solid year. Would that be sufficient for a dedicated student?

In no way do I expect to be a master within a year; hopefully I can know enough calculus and linear algebra to be well on my way with reading introductory physics texts on GR.

23. Aug 10, 2004

### mathwonk

Thank you Singleton for your kind response, I appreciate it very much.

As to how long it takes to master both volumes of Apostol, it could take a while. I have not been through them both myself, and my recommendation is based on one summer class in which I taught part way through volume 1, plus the testimonial of a mathematician friend who studied from Apostol as a young math major at MIT.

Apostol was written a little after my own education, when we used Courant. My son took a university course from volume 2, (without the background of volume 1).

It is probably standard to devote one year to each volume, in a university course, while pursuing other studies, and having lectures and homework. I have taught pretty much the whole of Spivak's Calculus book (comparable to Apostol vol. 1) in what to me is a world record 8 weeks, also to a class of returning teachers one summer.

I graded about 400 pages of homework a week for the whole class of about 10 students, and one of them told me they were spending up to 8 hours a day on the class. I recall the teacher of the other (abstract algebra) class they were taking complained they had no time left for his class. That summer was a wonderful experience for me.

I have also taught almost all the way through Spivak's (140 page) Calculus on Manifolds to a university class of seniors, covering most of it in either one semester or one year!, I forget. This class was very inexperienced and knew little or no linear algebra, which Spivak assumed.

I eventually came to prefer Apostol because of what I considered the slightly more scholarly presentation, but Spivak is written in a less dry, more fun way, and is probably preferred by most undergraduates. If you have access to a library, you might compare those two books, and perhaps also Courant, which also consists of two volumes.

I have all these books as permanent entries in my own library.

When teaching through Spivak's Calculus, and comparing with Courant, I came away with the impression Spivak had likely learned from Courant himself, and written his own book as a modernization, but this is a guess.

At any rate, with books of this quality, anything you get out of them is a plus, whether you get all the way through or not, so just begin one, and try not to give up or get bogged down too much. Fortunately you have this fine site to use as help source from the many expert respondents.

My impression from reading Spivak's Calculus on Manifolds is that the main theorems of advanced calculus are: 1) the inverse function theorem, 2) Fubini's theorem, 3) change of variables in integration, 4) Stokes theorem.

Thus if one learns those theorems one has essentially learned most of advanced calculus, whether one reads all the pages of a certain book or not. In general it may be better to think in terms of learning certain ideas or topics, rather then mastering certain books, or certain courses. A particular course or book may not do justice to a given idea but it is not necessarily the fault of the topic. In that case try another presentation.

I myself am soon about to return to teaching full time and will be less often here, but I have greatly enjoyed the atmosphere, and the stimulation of good interested questions from members like you, as well as the very helpful answers from many people. This site has definitely helped me get "up" for the start of the new semester.

Does this help?

best wishes

24. Aug 10, 2004

### singleton

Mathwonk,

Wow a year per volume(!) is significant but if that is what is needed then so be it.

I realize that I shouldn't focus too much on the "books" but rather topics and so forth but I figured (at the cost) I only wanted to invest in one set of books right now (even at the discounted cost of buying them second hand).

I have been thinking another set would be helpful though as I could get many views (perhaps I would appreciate certain topics described by one and other topics as described and presented by the second).

Thanks again!

25. Sep 29, 2008

### Ulagatin

Hi Mathwonk,

Brilliant posts throughout this forum. I am sure they have been very insightful to Singleton, but they have also been very insightful to myself, a third-party. Well done!

I understand that you are a mathematician, and don't do much, if anything to do with theoretical physics in your daily work as a maths professor, but if you could elaborate on the connections you mention here, that would be very much appreciated. I'd be very interested to hear about it, particularly in relation to string theory, as I think it is a fascinating topic!

Cheers,
-Davin