Geometrical books (differential geometry, tensors, variational mech.)

In summary: It's not terribly expensive, and the exposition is good.In summary, these are good books that focus on geometric interpretations. Most modern mathematics books lack these interpretations, and only consists of theorems and proofs. It seems that most modern mathematicians are pure left-brain sequential thinkers that do not have a lot of visualization capabilities.
  • #1
Idun
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I am looking for math books that focus on geometrical interpretations. Sadly most of the modern books lack these interpretations and only consists out of theorems and proofs. It seems to me that most modern mathematicians are pure left-brain sequential thinkers that do not have a lot of visualization capabilities.

I did some prior research on differential geometry, tensor calculus and variational mechanics and luckily Dover publishes really cheap but good books on these topics. My most recent purchases are:

-> Lectures on Classical Differential Geometry (Dirk Struik)
-> Lagrangian and Hamiltonian Mechanics (Calkin)
-> The variational principles of Mechanics (Cornelius Lanczos)
-> Vector and Tensor Analysis with Applications (Borisenko & Tarapov)

Some former threads mention Do Carmo but a quick glance in the ebook tells me it doesn't offer anything new.

Other books who are on my radar are "Geometry of Physics" by Theodore Frankel (hence why post is in this topic) and "Geometry, Topology and Physics" by Nakahara. Sadly I can't find pdf's of these to look into. (I always buy the books that I like, I just take precautions)

EDIT: I found Nakahara but for a geometry book, it doesn't contain a lot of geometrical pictures. I am looking for geometrical insights like for example Pythagoras visual proof:

gautam-geometry-proofs-07-1609747137.png
https://mathoverflow.net/questions/...ch-technical-detail-and-so-little-enlightenme

del.png
 
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  • #2
try david hilbert: geometry and the imagination, and david henderson: differential geometry, a geometric introduction.
 
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  • #3
I do recommend the Frankel book because of the careful pedagogy that's gone into it. As far as I can tell, there's not much compelling in the 2nd edition if you can find a cheaper copy of the 1st edition.

Tristan Needham's new "visual" book on differential geometry should be available soon. You can preview it on Amazon:

https://www.amazon.com/dp/0691203709/?tag=pfamazon01-20

Similar books in this vein are

Jon Pierre Fortney, A Visual Introduction to Differential Forms and Calculus on Manifolds
David Bachman, A Geometric Approach to Differential Forms 2nd ed.

The Bachman book is an easy read.

I'd also recommend studying the "classical" theory of curves and surfaces. I like the book by Millman & Parker, Elements of Differential Geometry.
 
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Related to Geometrical books (differential geometry, tensors, variational mech.)

1. What is differential geometry?

Differential geometry is a branch of mathematics that studies the properties of curves and surfaces using techniques from calculus and linear algebra. It is used to study the geometry of objects in higher dimensions and is an essential tool in fields such as physics, engineering, and computer graphics.

2. What are tensors in geometrical books?

Tensors are mathematical objects used to represent linear relationships between vectors and other geometric quantities. In differential geometry, tensors are used to describe the curvature and other properties of spaces. They are also used in variational mechanics to describe the dynamics of physical systems.

3. What is variational mechanics?

Variational mechanics is a mathematical framework used to study the behavior of physical systems by finding the path or trajectory that minimizes a certain quantity, such as energy or action. It is based on the principle of least action, which states that the path taken by a physical system is the one that minimizes the action.

4. How are geometrical books used in real-world applications?

Geometrical books, specifically those focused on differential geometry, tensors, and variational mechanics, are used in a variety of real-world applications. They are essential in fields such as physics, engineering, and computer graphics to model and understand the behavior of objects in higher dimensions. They are also used in the study of general relativity and other branches of theoretical physics.

5. What are the prerequisites for studying geometrical books?

A strong foundation in calculus, linear algebra, and mathematical analysis is necessary for studying geometrical books. Familiarity with abstract mathematical concepts and the ability to think in higher dimensions are also important. Some knowledge of physics and mechanics may also be beneficial, especially for understanding applications in variational mechanics.

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