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Discrete calculus

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Discrete Calculus by Peter Saveliev


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