# Calculus

This diagram shows an approximation to an area under a curve. Credit: Dubhe.

Calculus uses methods originally based on the summation of infinitesimal differences.

It includes the examination of changes in an expression by smaller and smaller differences.

## Mathematics

Main source: Mathematics

Mathematics is about numbers (counting), quantity, and coordinates.

Def. "[a]n abstract representational system used in the study of numbers, shapes, structure and change and the relationships between these concepts"[1] is called mathematics.

## Differences

Here's a theoretical definition:

Def. an abstract relation between identity and sameness is called a difference.

Notation: let the symbol ${\displaystyle \Delta }$ represent difference in.

Notation: let the symbol ${\displaystyle d}$ represent an infinitesimal difference in.

Notation: let the symbol ${\displaystyle \partial }$ represent an infinitesimal difference in one of more than one.

## Changes

Main sources: Abstractions/Changes and Changes

Def. "[s]ignificant change in or effect on a situation or state"[2] or a "result of a subtraction; sometimes the absolute value of this result"[2] is called a difference.

## Derivatives

Def. a result of an "operation of deducing one function from another according to some fixed law"[3] is called a derivative.

Let

${\displaystyle y=f(x)}$

be a function where values of ${\displaystyle x}$ may be any real number and values resulting in ${\displaystyle y}$ are also any real number.

${\displaystyle \Delta x}$ is a small finite change in ${\displaystyle x}$ which when put into the function ${\displaystyle f(x)}$ produces a ${\displaystyle \Delta y}$.

These small changes can be manipulated with the operations of arithmetic: addition (${\displaystyle +}$), subtraction (${\displaystyle -}$), multiplication (${\displaystyle *}$), and division (${\displaystyle /}$).

${\displaystyle \Delta y=f(x+\Delta x)-f(x)}$

Dividing ${\displaystyle \Delta y}$ by ${\displaystyle \Delta x}$ and taking the limit as ${\displaystyle \Delta x}$ → 0, produces the slope of a line tangent to f(x) at the point x.

For example,

${\displaystyle f(x)=x^{2}}$
${\displaystyle f(x+\Delta x)=(x+\Delta x)^{2}=x^{2}+2x\Delta x+(\Delta x)^{2}}$
${\displaystyle \Delta y=(x^{2}+2x\Delta x+(\Delta x)^{2})-x^{2}}$
${\displaystyle \Delta y/\Delta x=(2x\Delta x+(\Delta x)^{2})/\Delta x}$
${\displaystyle \Delta y/\Delta x=2x+\Delta x}$

as ${\displaystyle \Delta x}$ and${\displaystyle \Delta y}$ go towards zero,

${\displaystyle dy/dx=2x+dx=limit_{\Delta x\to 0}{f(x+\Delta x)-f(x) \over \Delta x}=2x.}$

This ratio is called the derivative.

## Partial derivatives

Let

${\displaystyle y=f(x,z)}$

then

${\displaystyle \partial y=\partial f(x,z)=\partial f(x,z)\partial x+\partial f(x,z)\partial z}$
${\displaystyle \partial y/\partial x=\partial f(x,z)}$

where z is held constant and

${\displaystyle \partial y/\partial z=\partial f(x,z)}$

where x is held contstant.

## Areas

Main sources: Spaces/Areas and Areas

In the figure on the right at the top of the page, an area is the difference in the x-direction times the difference in the y-direction.

This rectangle cornered at the origin of the curvature represents an area for the curve.

Notation: let the symbol ${\displaystyle \nabla }$ be the gradient, i.e., derivatives for multivariable functions.

${\displaystyle \nabla f(x,z)=\partial y=\partial f(x,z)=\partial f(x,z)\partial x+\partial f(x,z)\partial z.}$

## Curvatures

Main sources: Spaces/Curvatures and Curvatures

The graph at the top of this page shows a curve or curvature.

## Variations

Main sources: Spaces/Variations and Variations

Def. "a partial change in the form, position, state, or qualities of a thing"[4] or a "related but distinct thing"[4] is called a variation.

## Area under a curve

Consider the curve in the graph at the top of the page. The x-direction is left and right, the y-direction is vertical.

For

${\displaystyle \Delta x*\Delta y=[f(x+\Delta x)-f(x)]*\Delta x}$

the area under the curve shown in the diagram at right is the light purple rectangle plus the dark purple rectangle in the top figure

${\displaystyle \Delta x*\Delta y+f(x)*\Delta x=f(x+\Delta x)*\Delta x.}$

Any particular individual rectangle for a sum of rectangular areas is

${\displaystyle f(x_{i}+\Delta x_{i})*\Delta x_{i}.}$

The approximate area under the curve is the sum ${\displaystyle \sum }$ of all the individual (i) areas from i = 0 to as many as the area needed (n):

${\displaystyle \sum _{i=0}^{n}f(x_{i}+\Delta x_{i})*\Delta x_{i}.}$

## Integrals

Def. a "number, the limit of the sums computed in a process in which the domain of a function is divided into small subsets and a possibly nominal value of the function on each subset is multiplied by the measure of that subset, all these products then being summed"[5] is called an integral.

Notation: let the symbol ${\displaystyle \int }$ represent the integral.

${\displaystyle limit_{\Delta x\to 0}\sum _{i=0}^{n}f(x_{i}+\Delta x_{i})*\Delta x_{i}=\int f(x)dx.}$

This can be within a finite interval [a,b]

${\displaystyle \int _{a}^{b}f(x)\;dx}$

when i = 0 the integral is evaluated at ${\displaystyle a}$ and i = n the integral is evaluated at ${\displaystyle b}$. Or, an indefinite integral (without notation on the integral symbol) as n goes to infinity and i = 0 is the integral evaluated at x = 0.

## Theoretical calculus

Def. a branch of mathematics that deals with the finding and properties ... of infinitesimal differences [or changes] is called a calculus.

"Calculus [focuses] on limits, functions, derivatives, integrals, and infinite series."[6]

"Although calculus (in the sense of analysis) is usually synonymous with infinitesimal calculus, not all historical formulations have relied on infinitesimals (infinitely small numbers that are nevertheless not zero)."[7]

## Line integrals

Def. an "integral the domain of whose integrand is a curve"[8] is called a line integral.

"The pulsar dispersion measures [(DM)] provide directly the value of

${\displaystyle DM=\int _{0}^{\infty }n_{e}\,ds}$

along the line of sight to the pulsar, while the interstellar Hα intensity (at high Galactic latitudes where optical extinction is minimal) is proportional to the emission measure"[9]

${\displaystyle EM=\int _{0}^{\infty }n_{e}^{2}ds.}$

## Hypotheses

Main source: Hypotheses
1. Calculus can be described using set theory.

9. R. J. Reynolds (May 1, 1991). "Line Integrals of ne and ${\displaystyle n_{e}^{2}}$ at High Galactic Latitude". The Astrophysical Journal 372 (05): L17-20. doi:10.1086/186013. Retrieved 2013-12-17.