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Discrete Calculus. An Introduction

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Discrete Calculus. An Introduction by Peter Saveliev

Foreword: The stated goal of the text is discrete calculus. In particular, this is about the calculus of differential forms. The continuous counterpart is developed first because, typically, it is not a part of a calculus course. Meanwhile, we want the discrete

  • at least, to mimic the continuous, and,
  • ideally, to converge to the continuous.

There are still many gaps in the exposition...


Introduction: Why do we need differential forms?

  1. Algebra
    1. Vector spaces and modules
    2. Commutative diagrams
    3. Dual spaces
  2. Cubical complexes
    1. Cell decomposition of images
    2. Cubical complexes
    3. Boundary operator of cubical complex
    4. Cubical chain complex
  3. One-dimensional calculus in multi-dimensional spaces
    1. Differentials
    2. Vector and covector fields
    3. Motion and diffusion
    4. Modelling with discrete vector fields and forms
    5. Ordinary differential equations
    6. Partial differential equations
  4. Algebra
    1. Quotients of vector spaces
    2. Multivectors
    3. Multilinear forms
  5. Continuous differential forms
    1. Examples of differential forms
    2. Algebra of differential forms
    3. Wedge product of continuous forms
    4. Exterior derivative
    5. Properties of the exterior derivative
  6. Cubical differential forms
    1. Discrete differential forms
    2. Chains vs cochains
    3. Algebra of discrete differential forms
    4. Calculus of discrete differential forms
    5. Exterior derivative with Excel
    6. Calculus on chains
    7. Continuous vs discrete differential forms
  7. Integration of differential forms
    1. Orientation
    2. Integration of differential forms of degree 0 and 1
    3. Orientation of manifolds
    4. Integral theorems of vector calculus
    5. Integration of differential forms of degree 2
    6. Properties of integrals of differential forms
    7. General Stokes Theorem
  8. More algebra
    1. Hodge duality
    2. Inner product
    3. Tensor product
    4. Tensors
  9. Vector calculus and differential forms
    1. Forms vs vector fields and functions
    2. Identities of vector calculus
    3. Hodge duality of differential forms
    4. Hodge duality of cubical forms
    5. Discrete Hodge star operator
    6. Second derivative and the Laplacian
    7. Diffusion
  10. Differential geometry
    1. Geometry in calculus
    2. Metric tensor in dimensions 1 and 2
    3. Geometric Hodge duality
    4. Diffusion with various geometry
    5. Arc-length and curvature
    6. Hodge decomposition
    7. Lengths of digital curves
    8. Convergence of the discrete to the continuous
  11. Manifolds and differential forms
    1. Manifolds model a curved universe
    2. More about manifolds
    3. Tangent bundle
    4. Tangent bundles and differential forms
    5. Cubical tangent bundle
    6. Modelling motion on manifolds
    7. Configuration spaces
  12. Maps
    1. Examples of maps
    2. Change of variables for differential forms
    3. Cell maps
    4. Chain operators
  13. Partial Differential equations
    1. Darcy's flow
    2. Wave equation
    3. Navier–Stokes equations
    4. Maxwell equations
    5. Isotropy in numerical PDEs
  14. Tensor fields



Addendum



  1. Cubical homology
    1. Topology in real life
    2. Betti numbers
    3. Quotients of vector spaces
    4. Homology of cubical complexes
    5. Examples of homology of cubical complexes
    6. Realizations of cubical complexes
    7. Homology operators
  2. Cohomology
    1. Topology via Calculus
    2. Calculus and algebra vs topology
    3. Closed and exact forms
    4. Closedness and exactness of 1-forms
    5. Homotopy
    6. Simple connectedness
    7. de Rham cohomology
    8. Cochain complexes and cohomology
    9. Cohomology of figure 8