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This list of books provides the most direct and rigorous route to understanding differential geometry, the mathematical language of physics. Each selection thoroughly addresses its subject matter. The list does not need to be read linearly or only one book at a time. It is encouraged to go between books and/or read several together to acquire the necessary language and understand the motivations for each idea. The greatest hurdles are the motivation to learn and developing an understanding of the language of mathematics.

See the image on the right for a visual representation of its dependencies.

Also see this list of video lectures.

List Structure

The Royal Road to Differential Geometry and Physics is the list's core. While on that track, you should refer to the Fill in Gaps and Backbone sections as needed or as you desire.

The Fill in Gaps section covers the knowledge acquired in a strong high school mathematics education. Refer to it as needed, or begin there to develop your core skills.

The Backbone section supports the ideas in the Royal Road. Refer to it to strengthen your understanding of the ideas in the main track and to take those ideas further.

Fill in Gaps

Basic Mathematics

Review of arithmetic, algebra, trigonometry, logic, and geometry by Serge Lang.

Calculus

Overview of Calculus by Tom Apostol.

Royal Road to Differential Geometry and Physics

Sets for Mathematics

Categorical approach to set theory by F. William Lawvere.
Backbone reference:

• Set Theory and Metric Spaces
• Foundations of Analysis

Linear Algebra

Overview of linear algebra by Georgi Shilov.

Mechanics

Classical mechanics of physics by Lev Landau.
Prerequisite:

• Calculus

Backbone reference:

• Ordinary Differential Equations

The Classical Theory of Fields

Physics by Lev Landau.
Prerequisite:

• Linear Algebra

Tensor Analysis on Manifolds

Tensor analysis by Richard Bishop and Samuel Goldberg.
Prerequisite:

• Linear Algebra

Backbone reference:

• Principles of Mathematical Analysis
• Topology: A Categorical Approach

Lectures on Differential Geometry

Differential geometry by Shlomo Sternberg.
Prerequisite:

• Linear Algebra

Backbone reference:

• Principles of Mathematical Analysis
• Topology: A Categorical Approach

Cohomology & Differential Forms

Cohomology and differential forms by Isu Vaisman.
Backbone reference:

• Algebra: Chapter 0
• Algebra

Backbone

Set Theory and Metric Spaces

Set theory and metric spaces by Irving Kaplansky.

Foundations of Analysis

Analysis, intro to numbers, by Edmund Landau.

Principles of Mathematical Analysis

Mathematical analysis by Walter Rudin.

Ordinary Differential Equations

Ordinary differential equations by Vladimir Arnold.

Topology: A Categorical Approach

Topology by Tai-Danae Bradley, Tyler Bryson, Josn Terrilla.

Complex Analysis

Complex analysis by Lars Ahlfors.

Applications of Lie Groups to Differential Equations

Applications of Lie Groups to Differential Equations by Peter Olver.

Algebra Chapter 0

Algebra by Paolo Aluffi. Easier than Lang's, but less direct.

Algebra

Algebra by Serge Lang. The most direct approach to the subject.