Discrete Calculus by Peter Saveliev
Some of this material is now appearing as a part of a general purpose calculus book I am working on: Calculus Illustrated. The program is described here: From the discrete to the continuous. There is also an article “Discrete calculus” for Wikipedia.
Chapter 0: Freshman's introduction to discrete calculus
- Treat calculus discretely!
- Calculus I: functions of single variable
- Calculus II: parametric curves and ODEs
- Calculus III: functions of several variables and PDEs
Chapter 1: Calculus in dimension $1$
Chapter 2: Chains
Chapter 3: Calculus on chain complexes
Chapter 4: Geometry and calculus
- Metric tensor
- Metric complexes
- Diffusion equation and wave equation
- Convergence of the discrete to the continuous