Good Texts for Independent Study

In summary, Spivak's Calculus on Manifolds is a well-grounded introduction to the calculus of tensors and differential forms on manifolds, while Comprehensive Introduction to Differential Geometry by Spivak is the best overall diff geom book out there. If you have little geometric experience outside of high school geometry, then you might want to start out with Spivak's Volumes 1 and 2, which are easier.
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Hello.

I was wondeirng if anyone knew any good textbooks on Differential Geometry for independent study, at an undergraduate level. Thanks a bunch!

-Tim
 
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  • #2
Start with Spivak's Calculus on Manifolds for a well-grounded introduction to the calculus of tensors and differential forms on manifolds. Note that there is not a single useless or wasted exercise in the entire book. All of them should be taken as part of the text. You can then use Spivak's "Comprehensive Introduction to Differential Geometry".
 
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  • #3
although spivaks comprehensive text is the best overall diff geom book out there, it is so loong you can get lost or bogged down.


you might try some easier books like Noel J Hicks, or do Carmo, or Ted Shifrin's notes availablef ree from his website at University of georgia first.

Spivak spends an entire thick book, vol 1, on foundations of diff manifolds, a bit of a heavy meal. then vol 2 is a fantastic intro to the curvature tensor, the heart of diff geom. then it just goes on from there.

so if you get bored in vol 1 just go on to vol 2 for the geometry and refer back to vol 2 for needed facts. of course you need the definition of a manifold and a tangent space but that's about all.

a very easy book to read on a closely related subject diff top, is by guillemin and pollack, and intended for undergrads.
 
  • #4
If you have little geometric experience outside of high school geometry, then you might not want to start out with Spivak. Investigating 2- and 3-diml. differential geometry first, in order to build up some intuition particularly with regard to curvature, might be the best option... leaving small Spivak for, say, a senior thesis.

Thus, I would recommend John Oprea's Differential Geometry, Millman and Parker's text (Elements of Differential Geometry, I think), and/or Do Carmo's book on the geometry of curves and surfaces.
 
  • #5
well of course it depends on how strong the student is. I taught from little spivak in my senior level advanced calc course as a Pre PhD teacher at a small regional school 30 years ago, and it was too strong for the weaker ones and excellent for the strongest one. Our own state university uses it for junior level math majors and it works about right there. At Harvard they often start out strong sophomores and freshmen on much harder stuff.

So shop around among these recommendations for the one that lights your fire without scorching you.
 
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  • #6
mathwonk said:
although spivaks comprehensive text is the best overall diff geom book out there, it is so loong you can get lost or bogged down.


you might try some easier books like Noel J Hicks, or do Carmo, or Ted Shifrin's notes availablef ree from his website at University of georgia first.

Spivak spends an entire thick book, vol 1, on foundations of diff manifolds, a bit of a heavy meal. then vol 2 is a fantastic intro to the curvature tensor, the heart of diff geom. then it just goes on from there.

so if you get bored in vol 1 just go on to vol 2 for the geometry and refer back to vol 2 for needed facts. of course you need the definition of a manifold and a tangent space but that's about all.

a very easy book to read on a closely related subject diff top, is by guillemin and pollack, and intended for undergrads.



Could you please give me the website for the note?
 
  • #9
Flanders has a good book on diff G. My teacher tought me from it, I was pleased with its quality.
 

What is the purpose of independent study?

The purpose of independent study is to allow students to take control of their own learning and explore topics that interest them in depth. It requires self-motivation, time management, and critical thinking skills.

What makes a good text for independent study?

A good text for independent study should be engaging, challenging, and relevant to the topic being studied. It should also provide clear and thorough explanations, as well as opportunities for students to apply their learning.

How do I choose the right text for my independent study?

Choosing the right text for independent study requires careful consideration of your goals, interests, and abilities. It is important to find a text that you are truly interested in and that will challenge you in a meaningful way.

Can I use multiple texts for my independent study?

Yes, you can use multiple texts for your independent study. In fact, it is often beneficial to use a variety of resources to gain a well-rounded understanding of your topic. Just make sure that the texts you choose complement each other and do not overlap too much.

What strategies can I use to effectively study a text for independent study?

Some effective strategies for studying a text for independent study include creating a study schedule, taking notes, summarizing key points, and actively engaging with the material through discussions or reflections. It is also important to regularly review and reflect on your learning to ensure understanding and retention.

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