Regarding to Spivak's Differential Geometry trilogy

Click For Summary
SUMMARY

The discussion centers on the prerequisites for studying M. Spivak's first volume of his five-volume series on differential geometry. It is established that a solid understanding of vector calculus is beneficial, although not strictly necessary, for engaging with Spivak's work. The participant expresses a preference for nonlinear learning and mentions their current studies in Loomis/Sternberg and Hubbard/Hubbard, as well as familiarity with Spanier's Algebraic Topology and Lang's Algebra. The consensus suggests that while Spivak's first volume assumes knowledge of "Calculus on Manifolds," readers may still approach it without that prerequisite if they are willing to navigate the material independently.

PREREQUISITES
  • Understanding of vector calculus concepts, including directional derivatives and line integrals.
  • Familiarity with Spivak's "Calculus on Manifolds."
  • Knowledge of algebraic topology, particularly from Spanier's "Algebraic Topology."
  • Experience with differential geometry and topology basics.
NEXT STEPS
  • Study vector calculus in depth to strengthen foundational knowledge.
  • Read Munkres' "Topology" for a comprehensive understanding of topics related to Spivak's work.
  • Explore Spivak's "Calculus on Manifolds" to grasp the foundational concepts he builds upon.
  • Investigate Loomis/Sternberg's and Hubbard/Hubbard's texts for additional perspectives on differential geometry.
USEFUL FOR

Students and enthusiasts of mathematics, particularly those interested in differential geometry and topology, as well as educators seeking to guide learners through Spivak's complex material.

bacte2013
Messages
394
Reaction score
47
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.
 
Last edited:
Physics news on Phys.org
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.
 

Similar threads

Geometry Geometrical books (differential geometry, tensors, variational mech.)
  • · Replies 2 ·
Replies
2
Views
3K
Classical Source recommendation on Differential Geometry
  • · Replies 6 ·
Replies
6
Views
3K
Geometry Differential Geometry: Book on its applications?
  • · Replies 14 ·
Replies
14
Views
4K
Seeking "Reference" Textbooks in Mathematical Physics
  • · Replies 1 ·
Replies
1
Views
2K
Geometry Confusion about Differential Geometry Books
  • · Replies 11 ·
Replies
11
Views
3K
Geometry "The Geometry of Physics" - Theodore Frankel
  • · Replies 6 ·
Replies
6
Views
4K
Geometry Spivak's differential geometry vs calculus on manifolds
  • · Replies 6 ·
Replies
6
Views
5K
Calculus Does Apostol Calculus Volume 2 cover sufficient multivariate calculus?
  • · Replies 7 ·
Replies
7
Views
5K
Geometry Book Recommendations in Differential Geometry
  • · Replies 19 ·
Replies
19
Views
4K
  • Sticky
Other Collection of Free Online Math Books and Lecture Notes (part 1)
  • · Replies 16 ·
Replies
16
Views
12K