Discrete calculus

Differential calculus concerns incremental rates of change and the slopes of piece-wise linear curves.

Integral calculus concerns accumulation of quantities and the areas under piece-wise constant curves.

Discrete differential calculus is the study of the definition, properties, and applications of the difference quotient of a function.

Discrete integral calculus is the study of the definitions, properties, and applications of the Riemann sums.

For example, travelling a steady 50 mph for 3 hours results in a total distance of 150 miles.

This connection between the area under a curve and distance traveled can be extended to any irregularly shaped region exhibiting an incrementally varying velocity over a given time period.

The sum of all such rectangles gives the area between the axis and the piece-wise constant curve, which is the total distance traveled.

, the new function is defined at the points: The fundamental theorem of calculus states that differentiation and integration are inverse operations.

Such basic ideas as the difference quotients and the Riemann sums appear implicitly or explicitly in definitions and proofs.

However, the Kirchhoff's voltage law (1847) can be expressed in terms of the one-dimensional discrete exterior derivative.

The main contributions come from the following individuals:[1] The recent development of discrete calculus, starting with Whitney, has been driven by the needs of applied modeling.

An example of the use of discrete calculus in mechanics is Newton's second law of motion: historically stated it expressly uses the term "change of motion" which implies the difference quotient saying The change of momentum of a body is equal to the resultant force acting on the body and is in the same direction.

Commonly expressed today as Force = Mass × Acceleration, it invokes discrete calculus when the change is incremental because acceleration is the difference quotient of velocity with respect to time or second difference quotient of the spatial position.

Starting from knowing how an object is accelerating, we use the Riemann sums to derive its path.

In engineering, difference equations are used to plot a course of a spacecraft within zero gravity environments, to model heat transfer, diffusion, and wave propagation.

The discrete analogue of Green's theorem is applied in an instrument known as a planimeter, which is used to calculate the area of a flat surface on a drawing.

For example, it can be used to calculate the amount of area taken up by an irregularly shaped flower bed or swimming pool when designing the layout of a piece of property.

In the realm of medicine, calculus can be used to find the optimal branching angle of a blood vessel so as to maximize flow.

In nuclear medicine, it is used to build models of radiation transport in targeted tumor therapies.

[5] In signal processing and machine learning, discrete calculus allows for appropriate definitions of operators (e.g., convolution), level set optimization and other key functions for neural network analysis on graph structures.

: The difference (or the exterior derivative, or the coboundary operator) of the function is given by: It is defined at each of the above intervals; it is a

is a set of simplices that satisfies the following conditions: By definition, an orientation of a k-simplex is given by an ordering of the vertices, written as

A simplicial k-chain is a finite formal sum where each ci is an integer and σi is an oriented k-simplex.

A cubical complex is a set composed of points, line segments, squares, cubes, and their n-dimensional counterparts.

The elements of the individual vector spaces of a (co)chain complex are called cochains.

For cubical complexes, the wedge product is defined on every cube seen as a vector space of the same dimension.

The cup product operation satisfies the identity In other words, the corresponding multiplication is graded-commutative.

The Laplace operator represents the flux density of the gradient flow of a function.

For instance, the net rate at which a chemical dissolved in a fluid moves toward or away from some point is proportional to the Laplace operator of the chemical concentration at that point; expressed symbolically, the resulting equation is the diffusion equation.

The codifferential is the adjoint of the exterior derivative according to Stokes' theorem: Since the differential satisfies

Slope:
Constant velocity
The Riemann sum is measuring the total area of the bars, defined by , between two points (here and ).
A simplicial complex.
The boundary of a boundary of a 2-simplex (left) and the boundary of a 1-chain (right) are taken. Both are 0, being sums in which both the positive and negative of a 0-simplex occur once. The boundary of a boundary is always 0. A nontrivial cycle is something that closes up like the boundary of a simplex, in that its boundary sums to 0, but which isn't actually the boundary of a simplex or chain.