What are some recommended textbooks for covering various areas of mathematics?

[Dethrone]

I am looking for textbook recommendations for the topics listed below (and more). The books that I currently have are the ones that I found researching, and please let me know if they'll be helpful or if there's any redundancy in the material/list. I'm definitely open to suggestions too. (Cool)

Vector Calculus: Calculus on Manifolds (Spivak)
Abstract algebra: Abstract Algebra (Dummit and Foote)
Complex Analysis: Real and Complex Analysis (Rudin)
Real Analysis: Principles of Mathematical Analysis (Rudin)I want to cover as much math as I can...preferably as much as a math undergraduate for now because some of it isn't covered in my engineering program. If there are any areas of mathematics that I should look at /left out (I am definitely missing many), please let me know :D

 

[Euge]

Hi Rido12,

The only thing I criticize is your choice of text for complex analysis. Rudin's Real and Complex Analysis is a difficult graduate text, and I strongly recommend you hold off on this book until you've handled Rudin's basic text. For complex analysis, I recommend Theodore Gamelin's text.

 

[alyafey22]

I always recommend first course in complex analysis by Zill.
For abstract algebra the book by Gallian is very nice.

 

[Fantini]

For multidimensional real analysis (vector calculus) you can't go worst with Spivak. Use Duistermaat and Kolk's Multidimensional Real Analysis, both volumes. :) I also like Fraleigh's A First Course in Abstract Algebra. For basic real analysis look for Bernd Schröder's Mathematical Analysis: A Concise Introduction. If you want to go with something Rudin-style look for Karl Stromberg's Real Analysis.

 

[Dethrone]

Thanks for the suggestions, everyone!

A friend of mine recently suggested this roadmap into studying undergraduate mathematics, let me know if you think it should be a different order:

Topology (James Munkres) > Calculus on Manifolds > Differential Geometry > Differential Topology > Abstract Algebra > Algebraic Topology

 

[Fantini]

Is your friend a geometer by any chance? :) Without further information about what constitutes Calculus on Manifolds, Differential Geometry and Differential Topology it is hard to order them, since there is so much overlap. Are you assuming classical differential geometry? Calculus on Manifolds is also called Differential Topology. Most people do Differential Topology before any Differential Geometry.

 

[Dethrone]

I think he's just a first year student. But here is what he's suggesting and I am only taking it with a grain of salt:

Topology (James Munkres)
Calculus on Manifolds (Munkres or Spivak)
Differential Geometry
Lectures on Differential Geometry (Series on University Mathematics, Volume 1): Shiing-Shen Chern, W. H. Chen, K. S. Lam: 9789810241827: Amazon.com: Books
Differential Topology (An Introduction to Differentiable Manifolds and Riemannian Geometry)
Algebraic Toplogy (Munkres, again I think)

 

[Fantini]

My recommendations are:

Topology: A Taste of Topology, Volker Runde
Calculus on Manifolds: Multidimensional Real Analysis, volumes I and II - Duistermaat/Kolk
Differential Geometry: First Steps in Differential Geometry - Riemannian, Contact and Symplectic - Andrew McInerney.

 

[Fallen Angel]

Hi Rildo,

Abstract algebra and algebraic topology doesn't really require any knowing about differential geometry , calculus on manifolds or differential topology, so you can see that as a tree with the root on topology and two paths, you can choose the path you want.

 

[Prometheus1]

We should have MHB study group! (Happy)

 

[jiasyuen]

Can anyone actually suggest some good books for any topics for me? I've already finished 'Core Maths for Advanced Level' By L.Bostock and S.Chandler. I should move on to a higher level. Because I realize what I learn is just the basic of Mathematics.