Regarding to Spivak's Differential Geometry trilogy

In summary, the conversation is about the first volume of M. Spivak's five-volume set on differential geometry/topology and whether a thorough understanding of vector calculus is necessary before reading it. The speaker mentions their current readings of Loomis/Sternberg, Hubbard/Hubbard, Spanier's Algebraic Topology, and Lang's Algebra, as well as their background in Singer/Thorpe and Engelking. The conversation also mentions that Spivak assumes knowledge of his Calculus on Manifolds and that the speaker believes Munkres' book covers the same topics better.
  • #1
bacte2013
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I would like to begin my first exploration of the arts of differential geometry/topology with the first volume of M. Spivak's five-volume set in the different geometry. Is a thorough understanding of vector calculus must before reading his book? I read neither of his Calculus nor Calculus on Manifolds, but I just begun to read Loomis/Sternberg (quite exciting) and Hubbard/Hubbard. Unfortunately, my knowledge of vector calculus is quite shaky (only knows the definitions of topics like directional derivative and line integrals). Please let me know if Spivak's first volume builds directly upon his "Calculus on Manifolds". If the prerequisite of vector calculus is not strictly necessary, I would like to begin reading the first volume as I like to learn nonlinearly.

I am also currently reading Spanier's Algebraic Topology and Lang's Algebra. I acquired the topological background from Singer/Thorpe and Engelking.
 
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  • #2
Yes, he also states in the first book that he assume knoweldge of his Calculus on Manifolds.

I feel that a better book which covers the same topics as his Calculus on Manifolds is the book of Munkres.
 

What is the purpose of Spivak's Differential Geometry trilogy?

The purpose of Spivak's Differential Geometry trilogy is to provide a comprehensive and rigorous introduction to the mathematical study of curved spaces. It covers topics such as manifolds, tensors, differential forms, and Riemannian geometry, with the ultimate goal of understanding the geometric structure of our universe.

What background knowledge is required to understand Spivak's trilogy?

Spivak's Differential Geometry trilogy assumes a strong foundation in multivariable calculus, linear algebra, and basic topology. Familiarity with abstract algebra and differential equations may also be helpful, but is not essential.

What are some notable features of Spivak's writing style in this trilogy?

Spivak's writing style is known for being clear, concise, and rigorous. He emphasizes the use of precise mathematical language and logical reasoning, while also providing intuitive explanations and helpful examples. He also includes exercises and problems throughout the trilogy to reinforce key concepts and techniques.

What makes Spivak's trilogy a valuable resource for learning differential geometry?

Spivak's trilogy is highly regarded for its thoroughness and depth of coverage. It presents the material in a logical and intuitive manner, with a focus on developing a solid understanding of the subject rather than just memorizing formulas. The exercises and problems also provide ample opportunity for practice and application.

Are there any prerequisites for reading Spivak's trilogy?

Yes, as mentioned earlier, a good understanding of multivariable calculus, linear algebra, and basic topology is necessary. Some familiarity with abstract algebra and differential equations may also be helpful, but not required. It is recommended to have a strong mathematical background before attempting to read Spivak's trilogy.

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