In summary, the conversation revolved around the request for rigorous theoretical books on mathematics for each branch, particularly in the areas of Elementary Algebra, Elementary Geometry, Calculus, Linear Algebra, Real Analysis, and the possibility of adding more branches such as Topology, Complex Analysis, and Differential Geometry. The highlighted books mentioned in the conversation included Gelfand's Algebra and Functions & Graphs, Burnside's Theory of Equations, Euler's Analysis of the Infinite, Bellman's Introduction to Inequalities, Umbarger's Logarithms, Kiselev's Geometry, Lang's Geometry, Gelfand's Trigonometry and Method of Coordinates, Gutenmacher's Lines & Curves, Serge Lang's Basic Mathematics, Spiv
  • #1
Kalvino
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0
What are some rigorous theoretical books on mathematics for each branch of it? I have devised a fantastic list of my own and would like to hear your sentiments too.

Elementary Algebra:

Gelfand's Algebra
Gelfand's Functions & Graphs
Burnside's Theory of Equations
Euler's Analysis of the Infinite
Bellman's Introduction to Inequalities
Umbarger's Logarithms


Elementary Geometry:

Kiselev's Geometry
Lang's Geometry
Gelfand's Trigonometry
Gelfand's Method of Coordinates
Gutenmacher's Lines & Curves


Overview: Serge Lang's Basic Mathematics

Calculus:

Spivak's Calculus
Apostol's Calculus
Courant's Introduction to Calculus & Analysis
Simmons' Calculus with Analytic Geometry
Hubbard's Vector Calculus


Linear Algebra:

Lang's Introduction to Linear Algebra
Axler's Linear Algebra Done Right
Friedberg's Linear Algebra
Hoffman-Kunze's Linear Algebra
Roman's Advanced Linear Algebra


Real Analysis:

Binmore's Mathematical Analysis
Pugh's Real Mathematical Analysis
Folland's Real Analysis
McDonald's A Course in Real Analysis


You may make additions to my list or add more branches like Topology, Complex Analysis and Differential Geometry if you like, but remember; the books should focus on the "Why?" rather than the "How?" or in other words; should be highly theoretical. Books like Stewart's Calculus don't classify as being theoretical.
 
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  • #2
I wouldn't say Simmons is a rigorous book. Yes, it is a good book, however, it is very hand wavy.
 
  • #3
I am surprised that you do not have Walter Rudin's real analysis book listed.
 
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  • #4
introduction to ordinary differential equations by coddington , i would consider theoretical at the elementary level. Everything is proved, starts with complex numbers 1st page!
 
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  • #5
How about Tom Apostol's Mathematical Analysis and Paul Halmos' Finite Dimensional Vector Space"?
 

Related to Theoretical Books on Mathematics

1. What is the purpose of theoretical books on mathematics?

The purpose of theoretical books on mathematics is to explore and explain the underlying concepts and principles that govern mathematical theories and their applications. These books provide a deeper understanding of the subject and its significance in various fields.

2. How are theoretical books on mathematics different from practical books?

Theoretical books on mathematics focus on the abstract concepts and theories, while practical books provide step-by-step instructions and examples for applying those concepts in real-world situations. Theoretical books also tend to be more comprehensive and in-depth, while practical books may have a narrower scope and focus on specific applications.

3. Are theoretical books on mathematics only for advanced mathematicians?

No, theoretical books on mathematics can be beneficial for anyone interested in understanding the fundamental principles of mathematics. While some books may be more advanced and require a strong mathematical background, there are also introductory books that are accessible to a wider audience.

4. How can theoretical books on mathematics be useful in other fields?

Theoretical books on mathematics can provide a foundation for understanding and applying mathematical principles in various fields such as science, engineering, economics, and computer science. They can also help develop critical thinking and problem-solving skills that are valuable in many industries.

5. Can theoretical books on mathematics be used for self-study?

Yes, theoretical books on mathematics can be used for self-study. However, they may require more effort and time compared to practical books, as they often involve complex concepts and proofs. It is helpful to have a strong mathematical foundation and to work through the problems and exercises to fully understand the material.

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