Discrete calculus

Elementary Discrete Calculus by Peter Saveliev

How far we can go without limits?

Some of this material has appeared as a part of a "general purpose" calculus book of mine: Calculus Illustrated.

For now, I just picked 3 chapters from the first 3 volumes of Calculus Illustrated. Here is a PDF.

There is also an article “Discrete calculus” for Wikipedia.

Preface

Chapter 0: Freshman's introduction to discrete calculus

1. Treat calculus discretely!
2. Calculus I: functions of single variable
3. Calculus II: parametric curves and ODEs
4. Calculus III: functions of several variables and PDEs

Chapter 1: Calculus in dimension $1$

1. Discrete functions
2. Chains and cochains
3. Elementary ODEs
4. Elementary PDEs
5. Chain maps

Chapter 2: Chains

1. The cubical domain
2. Simplicial complexes
3. Chain complexes
4. Chain maps

Chapter 3: Calculus on chain complexes

1. Duality and cochains
2. Cubical calculus
3. ODEs of cochains and ODEs of chain maps

Chapter 4: Geometry and calculus

1. Metric tensor
2. Metric complexes
3. Diffusion equation and wave equation
4. Convergence of the discrete to the continuous

• Link: Spreadsheets
• Link: Discrete calculus: contributors