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About D. Books by D. Trivia About Differential Geom No trivia or quizzes yet. I even sketched out a plan form such a course to present to brilliant 10 year olds, assuming neither calculus, topology, linear algebra or even trig, which could be taught in the course. For more general differential geometry, there is a book by David Henderson, which attempts to teach an intuitive understanding of the ideas, curvature, parallel transport, holonomy But the basis of this suggestion is the hypothesis that concepts are more fundamental than techniques for computing them.

Since it seems that the most important concept in differential geometry is curvature, the first job is to convey an appreciation for curvature and its role in geometry. This can be done naturally in a very elementary setting. Only afterwards does it seem important to train someone in the means of computing it, i. It seems to me most courses smash many topics as a differential geometry course; it can be confusing for a beginner.

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As far as i am concerned, differential geometry should only contain the following so that it can be seen as advanced calculus :. The rest of the stuff like curvature, geodesic, connections, orientation and so on should be given as a second course Riemannian geometry.

I am sure things will be much simpler for us guys oriented to applying it to some practical problem. From my experience however one has to push as far as riemannian geometry and some group theory lie group, specifically to see much of the practical benefits of it. But who is to say what can be useful Sign up to join this community. The best answers are voted up and rise to the top.

Gauss, normals and fundamental forms - Differential Geometry 34 - NJ Wildberger

Home Questions Tags Users Unanswered. Asked 8 years, 6 months ago.

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Active 4 years, 8 months ago. Viewed 8k times. Vic with the idea that the course would be of interest to mechanical engineering students think Lagrangian and Hamiltonian classical mechanics , physics students relativity , and math students interested in manifold theory. I don't know to what extent other places are doing this but I imagine it's becoming a fairly typical story nowadays. It seems to me that the only prerequisites needed are linear algebra and vector calculus.

You don't even need analysis and topology which are needed mainly for proofs. But I don't know of any textbook that takes this approach. After all, the people you talked to benefit from hindsight, but would they have known to take the class?


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Science students in particular strike me on average as having very definite opinions about what courses they ought to take because they're the kind of folks who read ahead in the textbook and look stuff up online. Nevertheless, I know for a fact that they're not always right.

One may argue, though, it is not sufficiently elementary. Maybe we can convince Dick Palais to write another book?

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But it seems like a lot of the mathematics in do Carmo's Differential geometry of curves and surfaces would be doable by someone who has had the calculus sequence. I wouldn't recommend that book for such a course -- you'd have to tell the students to skip too many things, and as far as I remember it has no applications -- but this seems like one possible jumping-off point. I can't say anything more specific; my copy of do Carmo is on the other side of North America from me.

The course is divided in essentially four parts: Definition of manifolds as submanifolds in Euclidean space, diffeomorphisms and smooth maps, Sard's theorem and degree mod 2.

A First Course In Differential Geometry download

Curves and surfaces in space. Frenet frame, curvature, torsion of curves; isoperimetric inequality. First and second fundamental form, mean and Gaussian curvature. Calculus of variations, geodesics, minimal surfaces. More about curvature, leading up to Gauss-Bonnet. The only thing I would add to this are explicit examples starting with but going beyond the standard constant curvature spaces. Real, complex, and quaternionic projective spaces. Compact Lie groups. Homogeneous spaces. Things like that. I wouldn't say that it's preparation for Part III because we don't do abstract manifolds or even abstract Riemannian metrics everything is done by pullback from the embedding ; there is a Part III Differential Geometry course which starts again from scratch.

It's a pity, because I'm not sure how one is supposed to do, say, Lie theory without already knowing some differential geometry There's quite a bit of overlap between the two courses, in my opinion, and I feel quite strongly that Part II students should be given the definition of an abstract manifold if only because the Riemann Surfaces course gives such a definition and a Riemannian metric if students can understand it in the context of General Relativity, they can understand it here!

Pete L. But that's how one builds a strong geometric intuition. Do you use lecture notes or a book? Do you focus on giving people the tools to rapidly pick-up applications, or do you stroll through the zoo picking up what you need as you go along?