# Read

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

## Royal Road to Differential Geometry and Physics

### Sets for Mathematics

Categorical approach to set theory by F. William Lawvere.

**Backbone reference:**

### Mechanics

Classical mechanics of physics by Lev Landau.

**Prerequisite:**

**Backbone reference:**

### 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.

**Backbone reference:**

## Backbone

### Topology: A Categorical Approach

Topology by Tai-Danae Bradley, Tyler Bryson, Josn Terrilla. Click here for the Open Access version.

### Applications of Lie Groups to Differential Equations

Applications of Lie Groups to Differential Equations by Peter Olver.