Advice for Taking Differential Geometry in College

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Discussion Overview

The discussion revolves around the appropriateness of taking a 500-level Differential Geometry course for a high school senior currently studying Differential Equations. Participants share their insights on prerequisites, the difficulty of the course, and the importance of foundational knowledge in related subjects.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • Some participants emphasize the importance of the prerequisites, particularly multivariable calculus and linear algebra, for success in Differential Geometry.
  • One participant suggests considering an Analysis or Abstract Algebra course instead, indicating that Differential Geometry may be challenging without a solid foundation.
  • Another participant shares their experience of struggling with the material when rushing through it, advocating for a thorough understanding of concepts before advancing.
  • Concerns are raised about the appropriateness of a 500-level course for a high school student, with one participant expressing surprise that such a course is accessible at this educational level in the U.S.
  • Some participants recommend reviewing online resources related to the prerequisites to better prepare for the course.
  • There is a suggestion that grounding in tensors could be beneficial for understanding Differential Geometry, although the exact level of the course remains uncertain.

Areas of Agreement / Disagreement

Participants generally agree on the necessity of a strong foundation in the prerequisites for Differential Geometry, but there is no consensus on whether the original poster should take the course. Opinions vary on the appropriateness of the course level and the readiness of the student.

Contextual Notes

Some participants express uncertainty about the structure of algebra courses at the university and the implications of taking them at different institutions. The discussion also reflects differing educational standards and expectations between the U.S. and the U.K.

amcavoy
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I am a senior in HS right now. I am taking my classes at a local university because I finished my HS classes early, but will still graduate with my class this June. I am taking Differential Equations now, but would like to take a geometry class next semester. The prereqs are multivariable calculus, linear algebra and diffeq. However, it is a 500-level class (Differential Geometry). To those who have taken the class or are familiar with it: Would you recommend I take it? I would take an algebra class, but I'm not sure where I'm going to school next year so I don't want to take algebra I at this school and algebra II at another school; I'd rather take them at the same place. I appreciate your help.

Thanks a lot,

Alex
 
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I'm pretty positive that Differential Geometry is not easy, and that the pre-requisites are a must in this case.

Take an Analysis course or something. Or even Abstract Algebra.

I'm not sure what your goals are, but they seem to be geared towards General Relativity... maybe.

Take your time and challenge yourself on the current topics you are studying. Don't rush through it because it will catch up to you. Try to fully understand the concepts before moving on.

I tried going through Spivak's text independently before, but I tried to rush through it. After a couple chapters, I noticed my skills were crap and got discouraged, so I stopped. I now take my time and slowly move on. I give myself deadlines that are generous and that are obtainable. I'm doing very well right now. I take my time reading the chapter and fully absorbing everything he says because that's the only examples I get (independent remember). I also ask myself honestly if I'm ready to move on to the next chapter (after doing some questions), but if I don't feel 100%, I pick a difficult question that I must solve before moving on. Normally, after solving the difficult question I feel that I am prepared to move on.
 
Take your time and challenge yourself on the current topics you are studying. Don't rush through it because it will catch up to you. Try to fully understand the concepts before moving on.
I agree. Learning is not a race.
 
apmcavoy said:
I am a senior in HS right now. I am taking my classes at a local university because I finished my HS classes early, but will still graduate with my class this June. I am taking Differential Equations now, but would like to take a geometry class next semester. The prereqs are multivariable calculus, linear algebra and diffeq. However, it is a 500-level class (Differential Geometry). To those who have taken the class or are familiar with it: Would you recommend I take it? I would take an algebra class, but I'm not sure where I'm going to school next year so I don't want to take algebra I at this school and algebra II at another school; I'd rather take them at the same place. I appreciate your help.
Thanks a lot,
Alex
By the looks of what I've seen from differential geometry, multivariable calculus and linear algebra are very important pre-requisites.

By the way, if you're already at differential equations, why would you need to take algebra I and II? :\ Weren't you supposed to take those before Calc. I-II? Or is Linear Algebra split into two parts at your university?

Anyway, if you really want to go for differential geometry next semester, read some online texts on Calc. III, (There's a good book from MIT linked to in the links section) Linear Algebra and Differential Geometry itself before you decide you REALLY want to skip ahead to that.

Best of luck to you... and let us know how it goes. :)
 
Pseudo Statistic said:
By the looks of what I've seen from differential geometry, multivariable calculus and linear algebra are very important pre-requisites.
By the way, if you're already at differential equations, why would you need to take algebra I and II? :\ Weren't you supposed to take those before Calc. I-II? Or is Linear Algebra split into two parts at your university?
Anyway, if you really want to go for differential geometry next semester, read some online texts on Calc. III, (There's a good book from MIT linked to in the links section) Linear Algebra and Differential Geometry itself before you decide you REALLY want to skip ahead to that.
Best of luck to you... and let us know how it goes. :)
Oh yeah, by algebra I and II I mean modern algebra I and II (groups, rings, fields, etc.). I'll see how it goes.

Thanks a lot,

Alex
 
I'm feeling a little disgruntled/surprised that differential geometry is a high-school subject in the U.S. I'm sure it not taught until the 3rd year in a lot of half-decent universities (which is what mine is supposed to be) in the U.K. Maybe yours will be a lower level course though? I think GR was seen almost as a pre-requisite to mine. It's not the first time I've felt that U.S. education (in science at least) is more advanced though.
 
Mulder said:
I'm feeling a little disgruntled/surprised that differential geometry is a high-school subject in the U.S. I'm sure it not taught until the 3rd year in a lot of half-decent universities (which is what mine is supposed to be) in the U.K. Maybe yours will be a lower level course though? I think GR was seen almost as a pre-requisite to mine. It's not the first time I've felt that U.S. education (in science at least) is more advanced though.
It's not a high school class. I live right near a university that offers it, where I have been taking my classes past calculus (which is the last class my high school offers). I have to pay the required fees as if I were a regular student at the university, but I am able to take some classes there.

Alex
 
^Sorry I didn't really read your original post very well.

It sounds like a good opportunity, and one that isn't available to many, so you should consider it. I think some grounding in tensors is usually helpful to differential geometry, though I don't know how advanced a 500 level course is. If you're confident with calculus then maybe it's worth a go. I do agree though with other posters saying that diving into something before you're ready for it would be a waste of money. It's hard sometimes, but just be patient.
 

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