Studying Spivak: Calculus on Manifolds & Diff Geom, Worth It?

[SeReNiTy]

After studying Calculus by Spivak, I was enlightened by his writing style. Just wondering is his books "Calculus on Manifolds" and "Introduction to Differential Geometry Volume 1" worth purchasing if i plan to study diff geom?

 

[mathwonk]

calcuus on manifolds is a small readable book for a small price. it is highly recommended.

the 5 volume book on differential geometry is a classic, especially volume 2 which translates and explains riemanns own original treatment of the curvature tensor, there is nothing like it anywhwere else.

but it is up to you to decide if a 5 volume 2000-3000 page is worth it to you. at best it will take you years to read all of it. i suggest starting with volumes 1 and 2.

 

[mathwonk]

or look at it in the library.

 

[hypermorphism]

I have never found anything like Spivak (although Apostol and Courant are great for calculus as well). "Calculus on Manifolds" is pretty much the definitive treatment of vector calculus for those planning to study differential geometry. As said above, his comprehensive treatment of differential geometry is a classic and contains just as many insightful exercises and exacting yet intuitive definitions and theorems as his previous works, but is very long! :D Don't try to learn the subject from there unless you plan to take a few years. Get a shorter companion book to differential geometry as well. These volumes will then help you master the subject.

 

[mathwonk]

notice his comprehensive book really is comprehensive. it covers a lot besides diff geom proper. the whole first volume is on differentiable manifolds and related topics, including algebraic topology via differential forms, i.e. de rham theory, and i believe a tiny sample of lie groups.

he throws in problems with hints on useful topics like comoputing the dimension of avrious matrix groups. it is very sueful.

then volume 2 is the classic on gauss and riemanns work with modern explanatiions. this is actually as far as the course went. the next three volumes wre written afterwards.

vol 3 is a treatment of classical surfaces in 3 spoace i believe. 4 i don't know, and 5 is i believe on characteristic classes via forms, cherns original approach by the way, culminating in the general gauss bonnet theorem. there is also a section called a word from our sponsor on pde.