Thread: geometry book(s) recommendation

Hi there!

Can you recommend any good geometry books? I'm reading up on trigonometric identities, and think I need a little more geometry knowledge to better understand the identities...

Thanks!

I'm looking for some books on topology; does anybody have any recommendations?

What I'd particularly like the book(s) to cover is differential topology, analytic, algebraic, and geometric approaches to spaces and planes, Riemannian manifolds, metrics, and at least a brief outline of hyperboloid embedding.

Exercises featuring problems are not something I care for.

Originally Posted by Ellatha

I'm looking for some books on topology; does anybody have any recommendations?

What I'd particularly like the book(s) to cover is differential topology, analytic, algebraic, and geometric approaches to spaces and planes, Riemannian manifolds, metrics, and at least a brief outline of hyperboloid embedding.

Exercises featuring problems are not something I care for.

Some standard point-set topology books are

General Topology by Kelley

General Topology by Dugundji

Topology by Hocking and Young

Topology by Munkres

Originally Posted by rgba

Hi there!

Can you recommend any good geometry books? I'm reading up on trigonometric identities, and think I need a little more geometry knowledge to better understand the identities...

Thanks!

I have not personally read this book, but I think highly of the author, who is a very good algebraic geometer with a keen interest in primary and secondary education.

Geometry for the Classroom by C.H. Clemens

Though I am not a huge fan of on-screen texts, for those that are I recommend

http://uob-community.ballarat.edu.au...s/topology.htm for point set topology and

http://www.math.cornell.edu/~hatcher/AT/ATpage.html for the algebraic sort

For geometry, the written texts I like are:

Tensors, Differential forms and Variational Principles by Lovelock & Rund (Dover)

Tensor analysis on Manifolds by Bishop & Goldstein (also Dover)

Geometrical Methods of Mathematical Physics by B. Schutz

PS. Just remembered this as an online GR resource. Haven't read it, but a quick skim looks pretty good

Thanks to both of you.

Originally Posted by Guitarist

Though I am not a huge fan of on-screen texts, for those that are I recommend

http://uob-community.ballarat.edu.au...s/topology.htm for point set topology and

http://www.math.cornell.edu/~hatcher/AT/ATpage.html for the algebraic sort

For geometry, the written texts I like are:

Tensors, Differential forms and Variational Principles by Lovelock & Rund (Dover)

Tensor analysis on Manifolds by Bishop & Goldstein (also Dover)

Geometrical Methods of Mathematical Physics by B. Schutz

PS. Just remembered this as an online GR resource. Haven't read it, but a quick skim looks pretty good

Those geometry books are differential geometry. That is considerably beyond both high-school Euclidean geometry and trigonometry. Good books though, it you are ready for them.

But Ellatha specifically asked for texts on diff geom. I wasn't aware (s)he is a high school student. But, if I may say so, this attitude

Originally Posted by Ellatha

Exercises featuring problems are not something I care for.

is troubling. Learning mathematics is ALL about attempting exercises; they cannot be escaped if one wishes to understand the subject properly

Originally Posted by Guitarist

But Ellatha specifically asked for texts on diff geom. I wasn't aware (s)he is a high school student. But, if I may say so, this attitude

Originally Posted by Ellatha

Exercises featuring problems are not something I care for.

is troubling. Learning mathematics is ALL about attempting exercises; they cannot be escaped if one wishes to understand the subject properly

You are right -- on both counts.

rgba was asking for geometry as an aid to understanding trigonometry.

It also the case that if one needs basic topology, then Riemannin geometry is probably out of reach for the moment. And you can definitely forget about embeddings. Ditto for differential topology.

Guitarist,
I disagree. Exercise problems will develop a problem-solving ability within that particular branch of mathematics, but theorems are what truly allow you to understand it. Of course, I didn't say that I would mind problems being in the book, only that lack thereof would not be a problem.

As a note as well, I didn't say that I wished to read on differential geometry, I mentioned differential topology, which, while closely related, is a different branch of mathematics that combines the fundamental theorem of calculus with geometric principles. Later in life I may choose to learn more about the subject, however.

Originally Posted by Ellatha

Guitarist,
I disagree. Exercise problems will develop a problem-solving ability within that particular branch of mathematics, but theorems are what truly allow you to understand it.

But then, if you, yourself, can't prove a theorem, how do you know it is true?

Let me tell you a story. A while ago I had an irritating habit of vomiting stuff here that I had recently learned and found exciting. Some members here became interested, and asked questions. These guys (Wallaby & Chemboy in particular) became very dear friends precisely because they were willing to attempt the exercises I set.

Sometimes they got it right, sometimes not - it didn't mater, they were THINKING about the subject. I love them for that fact alone

And this is the key; learning mathematics at ANY level is not a passive process, you have to engage.

Originally Posted by Guitarist

But then, if you, yourself, can't prove a theorem, how do you know it is true?

I don't have to prove a theorem, I simply have to read the proof and understand what the author was doing.

Originally Posted by Guitarist

Let me tell you a story. A while ago I had an irritating habit of vomiting stuff here that I had recently learned and found exciting. Some members here became interested, and asked questions. These guys (Wallaby & Chemboy in particular) became very dear friends precisely because they were willing to attempt the exercises I set.

Sometimes they got it right, sometimes not - it didn't mater, they were THINKING about the subject. I love them for that fact alone

If I actively make a post stating that I'd like to independantly read up more on a particular branch of mathematics, than I'm obviously engaging myself.

Originally Posted by Guitarist

And this is the key; learning mathematics at ANY level is not a passive process, you have to engage.

If I told you that the product rule states that a derivative of a function is , and that a derivative is the slope of a tangent line at some point of a function of which can be used to find instantaneous rate of change (instantaneous velocity, for example), you could probably still find the instantaneous velocity of a function giving the height of a falling object at some time without me giving you a practice set of problems. It can be helpful to some people, but I don't find such trivial matters worth taking up my time. If I understand the concepts, than I can generally learn how to solve problems independantly. To "test" my knowledge I may sometimes set up problems for myself or do the exercises once, and if I don't get the percentage right that I'd like to, I'd simply reread the concepts again and than return and test myself again, and so on.

Just curious now, what are some good authors for Linear Algebra, Differential Equations, and Real Analysis texts?

For books on analysis, in my opinion the best is Professor Arthur Mattuck's book An Introduction to Analysis, and for differential equations and linear algebra I'm not so sure, as I learned differential equations in school rather than through a text, and haven't learned overally advanced linear algebra. I'm sure others will be able to help you here, though.

Originally Posted by Arcane_Mathematician

Just curious now, what are some good authors for Linear Algebra, Differential Equations, and Real Analysis texts?

At an undergraduate level:

Linear algebra: Halmos -- Finite Dimensinal Vector Spaces
Lang -- Linear Algebra


Differential equations: Boyce and Diprima -- Ordinary differential Equations
Braun -- Differential Equations and Their applications

Real analysis: Rudin -- Principles of Mathematical Analyis
Bartle -- Elements of Real Analysis[/i]

Originally Posted by Ellatha

If I actively make a post stating that I'd like to independantly read up more on a particular branch of mathematics, than I'm obviously engaging myself.

If you plan to just read and not try to work out the proofs for yourself you are probably just kidding yourself.

You have indicated interest in topics that are WAY beyond the background that you say you have. You have no chance of simply picking up book on, say,
Riemannan geometry and actually understanding it, even if you think you understand it.

You need to be able to construct the proofs of basic mathematics -- algebra, topology and analysis almost without thinking before you tackle the harder stuff.

For instance you have said that you have not studied linear algebra at an advanced level. But without being able to do linear algebra proofs in your head, you will not understand multilinear algebra that is necessary for differential forms which is necessary for understanding differential manifolds at a simple level which is necessary in order to even begin to tackle Riemannian geometry.

A tangent vector field is a derivation on the ring of smooth functions on a manifold. It that sounds foreign because of lack of familiarity with algebra, and it probably does, then you need to learn some basics, and to do that you need to do many proofs yourself.

I don't think that I ever said that I'd simply read and not work out proofs, but only that I don't have a need for trivial problems. I don't think I've ever given anything approaching a background of myself to anybody online besides a simple statement regarding linear algebra (I do understand matrices, determinants, vectors, and finding the solutions to higher order equations, just not the likes of Gaussian elimination). But, I suppose next time I could just use Amazon or equivalent if takes offense to others.

Originally Posted by Ellatha

I don't think that I ever said that I'd simply read and not work out proofs, but only that I don't have a need for trivial problems. I don't think I've ever given anything approaching a background of myself to anybody online besides a simple statement regarding linear algebra (I do understand matrices, determinants, vectors, and finding the solutions to higher order equations, just not the likes of Gaussian elimination). But, I suppose next time I could just use Amazon or equivalent if takes offense to others.

You won't get knowledgable recommendations at Amazon.

To make appropriate recommendations it is necessary to asses the background of the person asking. Your statement above is quite helpful and confirms my earlier impressions.

It is not that hard to figure out someone's background in mathematics if you have spent quite a bit of time with subject, including university teaching.

Good mathematics books, at a reasonable level, do not have trivial problems. If you find a problem that is trivial you should be able to do it in your head and move on.

Very well than -- I will respect your statements on the subject and read lower level mathematics in that case. But, as I give you my respect of somebody who's older more knowledgeable on the subject than me, I expect you to also not abridge my learning through your recommendations by the same token.

Originally Posted by Ellatha

Very well than -- I will respect your statements on the subject and read lower level mathematics in that case. But, as I give you my respect of somebody who's older more knowledgeable on the subject than me, I expect you to also not abridge my learning through your recommendations by the same token.

The books that were recommended are good books. They can be handled by anyone with the requisite "mathematical maturity" and do not require any specific pre-requisites. But you do need to study them actively and not passively.

They are not what I would call "low level". They are real mathematics. They just don't require a lot of background. In fact they will give you the background that you need to study more advanced mathematics.