Stokes' theorem relates the integral of a differential (n − 1)-form ω over the boundary ∂M of an n-dimensional manifold M to the integral of dω (the exterior derivative of ω, and a differential n-form on M) over M itself:

${\displaystyle \int _{M}\mathrm {d} \omega =\int _{\partial M}\omega .}$ 

One could think of differential k-forms as linear operators that act on k-dimensional "bits" of space, in which case one might prefer to use the bracket notation for a dual pairing. In this notation, Stokes' theorem reads as

${\displaystyle \langle \mathrm {d} \omega \mid M\rangle =\langle \omega \mid \partial M\rangle .}$ 

In finite element analysis, the first stage is often the approximation of the domain of interest by a triangulation, T. For example, a curve would be approximated as a union of straight line segments; a surface would be approximated by a union of triangles, whose edges are straight line segments, which themselves terminate in points. Topologists would refer to such a construction as a simplicial complex. The boundary operator on this triangulation/simplicial complex T is defined in the usual way: for example, if L is a directed line segment from one point, a, to another, b, then the boundary ∂L of L is the formal difference b − a.

A k-form on T is a linear operator acting on k-dimensional subcomplexes of T; e.g., a 0-form assigns values to points, and extends linearly to linear combinations of points; a 1-form assigns values to line segments in a similarly linear way. If ω is a k-form on T, then the discrete exterior derivative dω of ω is the unique (k + 1)-form defined so that Stokes' theorem holds:

${\displaystyle \langle \mathrm {d} \omega \mid S\rangle =\langle \omega \mid \partial S\rangle .}$ 

For every (k + 1)-dimensional subcomplex of T, S.

Other operators and operations such as the discrete wedge product,^[2] Hodge star, or Lie derivative can also be defined.

• Discrete differential geometry
• Discrete Morse theory
• Topological combinatorics
• Discrete calculus

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• Hirani Thesis on Discrete Exterior Calculus
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• Convergence of discrete exterior calculus approximations for Poisson problems, E. Schulz & G. Tsogtgerel, Disc. Comp. Geo. 63(2), 346 - 376, 2020
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