I do math
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.
A further set of texts extending this one, but working with the same basics has been produced leading all the way up and through gauge field theory, quantum mechanics, algebraic geometry, and quantum field theory here.
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 single and multi-variable calculus with applications to differential equations and probability by Tom Apostol.
Royal Road to Differential Geometry and Physics
Sets for Mathematics
Categorical approach to set theory by F. William Lawvere.
Backbone reference:
Linear Algebra
Linear algebra of linear equations, maps, tensors, and geometry by Georgi Shilov.
Mechanics
Classical mechanics of particles by Lev Landau.
Prerequisite:
Backbone reference:
The Classical Theory of Fields
Classical field theory of electromagnetism and general relativity by Lev Landau.
Prerequisite:
Tensor Analysis on Manifolds
Tensor analysis by Richard Bishop and Samuel Goldberg.
Prerequisite:
Backbone reference:
Lectures on Differential Geometry
Differential geometry by Shlomo Sternberg.
Prerequisite:
Backbone reference:
Cohomology & Differential Forms
Cohomology and differential forms by Isu Vaisman. Sheaf theoretic description of the cohomology of real, complex, and foliated manifolds.
Backbone reference:
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. Click here for the Open Access version.
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.