Discrete calculus article

Discrete calculus or "the calculus of discrete functions", is the mathematical study of incremental change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. The word calculus is a Latin word, meaning originally "small pebble"; as such pebbles were used for calculation, the meaning of the word has evolved and today usually means a method of computation. Meanwhile, calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the study of continuous change.

It has two major branches, differential calculus and integral calculus. Differential calculus concerns incremental rates of change and the slopes of discrete curves. Integral calculus concerns accumulation of quantities and the areas under and between such curves. These two branches are related to each other by the fundamental theorem of discrete calculus.

These concepts of change start in their discrete form. Then by making the increment smaller and smaller, we find their contonuous counterparts via this limit: $$\newcommand{\ra}[1]{\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!} \begin{array}{ccccc} \begin{array}{|cc|}\hline\text{ discrete }\\ \text{ calculus }\\ \hline\end{array}& \ra{\quad\Delta x\to 0\quad} &\begin{array}{|cc|}\hline\text{ infinitesimal }\\ \text{ calculus }\\ \hline\end{array} \end{array}$$

History

The early history of discrete calculus is the history of calculus.

Discrete calculus remain interlinked with infinitesimal calculus especially differential forms. Discrete calculus relies on "discrete differential forms", i.e., cochains. It cannot then be separated from the rest of exterior calculus or from algebraic topology. Therefore, the credit for the creation of discrete calculus should first go to the following individuals (roughly 1850 - 1950):

• Hermann Grassmann: exterior algebra
• Gregorio Ricci-Curbastro, Tullio Levi-Civita: tensor calculus
• Henri Poincaré: triangulations (barycentric subdivision, dual triangulation), Poincare duality, Poincare lemma, the first proof of the general Stokes Theorem, and a lot more
• Vito Volterra: "The first mathematician to have written down Stokes' formula for an arbitrary dimension was probably V. Volterra." -- Dieudonne
• L. E. J. Brouwer: simplicial approximation
• Élie Cartan, Georges de Rham: the notion of differential form, the exterior derivative as a coordinate-independent linear operator, exactness/closedness of forms, the de Rham's theorem (the de Rham cohomology is equivalent to the singular cohomology)
• Emmy Noether, Heinz Hopf, Leopold Vietoris, Walther Mayer: modules of chains, the boundary operator, chain complexes
• J.W. Alexander, Solomon Lefschetz, Lev Pontryagin, Andrey Kolmogorov, Norman Steenrod, Eduard Čech: the early cochain notions
• W. V. D. Hodge: the Hodge star operator, the Hodge decomposition
• Samuel Eilenberg, Saunders Mac Lane, Norman Steenrod, J.H.C. Whitehead: the rigorous version of algebraic topology
• Hassler Whitney: cochains as integrands

The current development of discrete calculus is driven by the needs of applied modeling.

Principles

Differential calculus is the study of the definition, properties, and applications of the difference quotient of a function. The process of finding the difference quotient is called differentiation. Given a function and a point in the domain, the difference quotient at that point is a way of encoding the small-scale (i.e., from the point to the next) behavior of the function. By finding the difference quotient of a function at every point in its domain, it is possible to produce a new function, called the difference quotient function or just the difference quotient of the original function. In formal terms, the difference quotient is a linear operator which takes a function as its input and produces a second function as its output. This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function is given the input three, then it outputs six, and if the squaring function is given the input three, then it outputs nine. The derivative, however, can take the squaring function as an input. This means that the derivative takes all the information of the squaring function—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to produce another function. The function produced by deriving the squaring function turns out to be the doubling function.

In more explicit terms the "doubling function" may be denoted by Template:Math and the "squaring function" by Template:Math. The "difference quotient" now takes the function Template:Math, defined by the expression "Template:Math", as an input, that is all the information—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to output another function, the function Template:Math, as will turn out.

The most common symbol for a difference quotient is: $$\frac{\Delta f}{\Delta x}.$$ This notation is known as Leibniz's notation.

If the input of the function represents time, then the difference quotient represents change with respect to time. For example, if Template:Math is a function that takes a time as input and gives the position of a ball at that time as output, then the derivative of Template:Math is how the position is changing in time, that is, it is the velocity of the ball.

If a function is linear (that is, if the graph of the function is a straight line), then the function can be written as Template:Math, where Template:Math is the independent variable, Template:Math is the dependent variable, Template:Math is the y-intercept, and:

[math]m= \frac{\text{rise}}{\text{run}}= \frac{\text{change in } y}{\text{change in } x} = \frac{\Delta y}{\Delta x}.[/math]

This gives an exact value for the slope of a straight line. If the graph of the function is not a straight line, however, then the change in Template:Math divided by the change in Template:Math varies. The difference quotient give an exact meaning to the notion of change in output with respect to change in input. To be concrete, let Template:Math be a function, and fix a point Template:Math in the domain of Template:Math. Template:Math is a point on the graph of the function. If Template:Math is the increment of $x$, then Template:Math is the value of $x$ after (or before) Template:Math. Therefore, Template:Math is the increment of Template:Math. The slope between these two points is

[math]m = \frac{f(a+h) - f(a)}{(a+h) - a} = \frac{f(a+h) - f(a)}{h}.[/math]

So Template:Math is the slope of the line between Template:Math and Template:Math.

Here is a particular example, the difference quotient of the squaring function at the input 3. Let Template:Math be the squaring function.

File:Sec2tan.gif

The difference quotient Template:Math of a curve at a point is the slope of the line tangent to that curve at that point. We considering the slopes of line. Here the function involved (drawn in red) is Template:Math. The tangent line (in green) which passes through the point Template:Nowrap has a slope of 23/4. Note that the vertical and horizontal scales in this image are different.

[math]\begin{align}f'(3) &={(3+h)^2 - 3^2\over{h}} \\ &={9 + 6h + h^2 - 9\over{h}} \\ &={6h + h^2\over{h}} \\ &= (6 + h) \end{align} [/math]