What Should You Study Before Spivak's Calculus on Manifolds?

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Discussion Overview

The discussion revolves around the preliminary knowledge and texts recommended before studying Spivak's "Calculus on Manifolds." Participants explore the necessary background in calculus, real analysis, and linear algebra, as well as the relationship between these subjects and the content of Spivak's work.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants suggest that knowledge of real analysis (e.g., Rudin) and linear algebra is beneficial before tackling Spivak's text.
  • Others argue that real analysis is not strictly necessary, emphasizing that Spivak's book serves as an introduction to multivariate calculus using modern methods.
  • One participant mentions that a strong foundation in calculus and linear algebra is sufficient, recommending Spivak or Apostol for calculus preparation.
  • There is a discussion about the differences in focus between Spivak and Rudin, with some noting that Spivak's work is more geometric while Rudin's is more analytic.
  • Several participants express interest in the progression of study after Spivak, with suggestions including Courant and Apostol for further analysis.
  • One participant reflects on their own background in calculus and analysis, questioning the value of revisiting calculus after studying analysis.
  • Another participant suggests that mastering analysis may enhance one's understanding of calculus, but does not assert this as a definitive conclusion.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of real analysis before studying Spivak's text, indicating that multiple competing perspectives remain. There is no consensus on a single recommended path of study.

Contextual Notes

Participants note that calculus and analysis are related but distinct fields, with varying degrees of overlap in the texts discussed. The discussion reflects differing opinions on the adequacy of various preparatory texts and the nature of the knowledge required.

Who May Find This Useful

This discussion may be useful for students considering the study of advanced calculus and analysis, particularly those interested in the transition from introductory calculus to more rigorous mathematical texts.

Winzer
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What are some preliminary texts/knowledge before approaching: Spivak's Calculus on Manifolds?
 
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Some real analysis (rudin) and linear algebra?
 


I don't think you need real analysis, even though some of Spivak's problems are harder than in Rudin. Spivak's manifolds is an introduction to multivariate calculus using modern methods, which is why it may be confused as a mutlivariable real analysis book. The only pre-requisite is a strong book in Calculus (see Spivak or Apostol) and linear algebra (see Axler). Spivak is not a full fledged real analysis book in multivariables, it is an introduction to the moderm techniques. So Rudin should come after.
 


I am using Spivak's regular book on Calculus right now. Would I be able to tackle it(Spivak's Calculus on Manifolds) afterwards?

What I am really trying to get at is sound knowledge in analysis. I heard that Spivak's Calculus and Calculus of Manifolds were excellent starters. I have heard that Rudin's text are subpar compared to most analysis text. What would come after Spivak? Courant? Apostol(Mathematical Analysis)?
 


spivak is a differential geometer, and rudin is an analyst. thus spivaks book is better for the geometric aspect and rudins is more precise for the analytic one.

e.g. you are better off learning about integration of differential forms from spivak, and maybe some other things (lebesgue integration?) from rudin.
 


Winzer said:
I am using Spivak's regular book on Calculus right now. Would I be able to tackle it(Spivak's Calculus on Manifolds) afterwards?

What I am really trying to get at is sound knowledge in analysis. I heard that Spivak's Calculus and Calculus of Manifolds were excellent starters. I have heard that Rudin's text are subpar compared to most analysis text. What would come after Spivak? Courant? Apostol(Mathematical Analysis)?

Calculus and analysis are not exactly the same thing. Calculus is more like elementary analysis. Books like Spivak, Courant, and Apostol teach you calculus, although do to their depth there will be a lot of overlap with analysis courses. Spivak's books in particular will prepare you for analysis, not make you master it. After Spivak's books, it is time to move onto real analysis which is more general. For that, the classic text is Rudin, "Principles of Mathematical Analysis". Another book which I am currently using is Pugh's "Real Mathematical Analysis", a book that is quickly becoming the new standard.
 


I was taught calculus from stewart, abeit more intensely from lectures but that was what the course was based on. I have since been learning analysis from rudin, is it worthwhile to go back and get better at calculus?

I think this adds to the discussion of the current thread.
 


i think you are all right, but you may enjoy browsing in a top calculus book like spivak, apostol, or kitchen.
 


Thank you all for the clarification. So mastering analysis would make one a master at calculus?

I guess I will start with Rudin's analysis book. What comes after that?
 

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