Trying to make sense of math sequence

[timeevaporate]

Hi All,

I posted this text below in another forum, but didn't get much feedback. I realize I might sound extremely ignorant, but I need directions. Thanks.


"Awhile ago I decided I wanted to learn Relativity comprehensively. I was told that I better brush up on my Differential Geometry if I wanted to delve deeper into Relativity. So I decide to find out what the prerequisites for Diff. Geo were. Some say, Calc 1-3 and Linear Algebra would suffice, others say I should take intro to Multilinear Algebra, whose prereqs are, apparently, Functional Analysis and Topology according to some; yet others claim I must know Linear Algebra and Abstract Algebra as a bridge to Multilinear Algebra and, further into, Diff. Geo. Then, I find out that Functional Analysis has its own prereq like Measure Theory. How do Real Analysis and Complex Analysis, Differential Equations fit into all this? Are there overarching/ more fundamental subjects that treat the above-mentioned subjects as their integral topics?

At this point I don't even care to know what EXACTLY I need for Diff Geo., because I'll, probably, be correctly told to ditch this and pick up that on the road to Diff. Geo, but now that I know these subjects exist I'd rather study them since I find the idea of them very interesting. And if it takes me forever to converge the knowledge of these subjects into a comprehensive intro into Relativity- so be it. I am not even worried about the result here, I want to enjoy the process since math/physics is a little hobby of mine and I am a late- commer to these subjects. I am not out to make a profession out of this, nor do I care to prove anything to anyone else. I feel just because I am a layman doesn't mean I have to limit myself to some very unsatisfying pop-science texts. So far, I am getting ready to teach myself Calc 3 and probably take an intro to Proofs.

That said, could somebody, please, weave these subjects and fit them with each other in a more logical and sequential way with, possibly, some missing subjects as prereqs and stepping stones for me?

By the way, a couple of very stupid questions- Is Analysis arranged in a manner Calculus is typically arranged? That is, Analysis 1, Analysis 2 etc? I mean Calc is self-contained, is Analysis also? If not, what would be the most important Analysis topics/ideas/concepts for people like me? Again, I am not scared of difficulties and I have my whole life to learn all this.

Thanks."

 

[halo31]

You seem quite confused. For general relativity you must know differential geometry and differential manifolds. But if what your asking for is what background you need and in what order? A typical order would be Calculus 1-3, differential equations, 1st semester linear algebra(maybe a second second semester course), Introduction to Proofs,Analysis 1 and 2, abstract algebra, General Topology. From here you could branch out into differential geometry etc. I wouldn't think you would need complex analysis or functional analysis but it doesn't hurt.

 

[timeevaporate]

You seem quite confused. For general relativity you must know differential geometry and differential manifolds. But if what your asking for is what background you need and in what order? A typical order would be Calculus 1-3, differential equations, 1st semester linear algebra(maybe a second second semester course), Introduction to Proofs,Analysis 1 and 2, abstract algebra, General Topology. From here you could branch out into differential geometry etc. I wouldn't think you would need complex analysis or functional analysis but it doesn't hurt.

Thanks for the response. The thing is I am not really all that interested in Relativity anymore. I guess I should have worded this differently. Right now I like the geometric part of Calculus a lot and anything that has to do with studying and classifying numbers- no so much. What would be the better way to study further if I am really into Calculus/Geometry and less into work with numbers? In other words, I'd like to know if there's some kind of ultimate Geometry challenge I could work my way up to. :)

Thanks.edit: Whatever the case, I think what you outlined seems like a good path to take, so I think I will go by it. Couple questions: what is the difference between first and second semester Linear Algebra courses? And what's the difference between Analysis 1 and 2? Thanks.

 

[halo31]

Oh 1st semester linear algebra is more computational then theoretical and it introduces you to the concepts of linear algebra like determinants, matrix theory, vector spaces,etc. Second semester linear algebra is all that but goes in more depth theoretically. What I meant by analysis 1 and 2(it goes by different names) is you basically cover proofs dealing with sequences all the way up to integration and differentiation in one dimension. In Analysis 2 you basically cover proofs involving calculus in several dimensions. Oh I see you like geometry. Well a good book that has both Calculus and Analytical geometry mixed would be by Calculus with Analytic Geometry by George Finlay Simmons. A good if your starting calculus. From there it You can go into Differential Geometry, Topology, Manifolds, Euclidean Geometry, Algebraic Geometry, etc. List goes on but like I said earlier if you have a good math foundation you could study whatever you have interest in.

 

[Matrix0]

Dear poster,

Thank you for reaching out and expressing your interest in learning about Differential Geometry and its prerequisites. It is completely normal to feel overwhelmed and unsure of where to start when delving into a new subject, especially one as complex as mathematics. Let me try to provide some guidance and clarification for you.

Firstly, it is important to note that there is no one set path to learning about Differential Geometry or any other subject in mathematics. Different universities and individuals may have different opinions on the necessary prerequisites or the best way to approach the subject. However, there are some common themes and concepts that are important to understand before diving into Differential Geometry.

As a general rule of thumb, it is important to have a strong foundation in calculus, linear algebra, and basic abstract algebra before attempting to tackle Differential Geometry. These topics are typically covered in calculus courses (Calc 1-3) and linear algebra courses. You may also encounter some differential equations in your studies, so having a basic understanding of that topic would be beneficial as well.

In terms of more advanced topics like Functional Analysis and Topology, they are not necessarily required for understanding Differential Geometry, but they do provide useful tools and concepts that can enhance your understanding of the subject. However, it is not necessary to have a deep understanding of these topics before starting your studies in Differential Geometry. It may be best to focus on the basics first and then delve into these more advanced topics as you progress.

Regarding your question about Analysis, it is not typically arranged in a manner similar to calculus. Analysis is a broad field that encompasses topics like real analysis, complex analysis, and functional analysis. These topics are not necessarily arranged in a sequential order, but rather build upon each other and can be studied in parallel. However, for someone with a general interest in mathematics, it may be beneficial to focus on real analysis and complex analysis first before delving into functional analysis.

In summary, my suggestion would be to focus on building a strong foundation in calculus, linear algebra, and basic abstract algebra before moving on to Differential Geometry. As you progress, you may encounter other topics that could enhance your understanding, but it is not necessary to have a deep understanding of all these topics before starting your studies in Differential Geometry. And most importantly, enjoy the process and don't be afraid to ask for help or clarification when needed. Good luck on your journey of learning!