# Dictionary Definition

calculus

### Noun

1 a hard lump produced by the concretion of
mineral salts; found in hollow organs or ducts of the body; "renal
calculi can be very painful" [syn: concretion]

3 the branch of mathematics that is concerned
with limits and with the differentiation and integration of
functions [syn: the
calculus, infinitesimal
calculus] [also: calculi (pl)]

# User Contributed Dictionary

## English

### Etymology

calculus, pebble (used for counting), diminutive of calx, limestone, + diminutive suffix -ulus### Noun

- Any formal system in which symbolic
expressions are manipulated according to fixed rules.
- vector calculus

- Differential calculus and integral calculus considered as a single subject.
- A stony concretion that forms in a
bodily organ.
- renal calculus ( = kidney stone)

- (dentistry) Deposits of calcium phosphate salts on teeth.

#### Synonyms

#### Derived terms

- calculus of sums and differences
- calculus of variations
- dental calculus
- differential calculus
- finite-difference calculus
- infinitesimal calculus
- integral calculus
- join calculus
- lambda calculus
- pi-calculus
- precalculus
- predicate calculus
- tensor calculus
- tuple calculus

#### Related terms

#### Translations

formal mathematical system

differential calculus and integral calculus
considered as a single subject

stony concretion in an organ

deposits on teeth

- German: Zahnstein
- Italian: tartaro, tartaro dentario
- Japanese: 歯石
- Polish: kamień nazębny
- Spanish: sarro

### See also

## Latin

### Noun

calculus m (plural calculi)#### Derived terms

# Extensive Definition

Calculus (Latin, calculus, a
small stone used for counting) is a branch of mathematics that includes
the study of limits,
derivatives, integrals, and infinite
series, and constitutes a major part of modern university
education. Historically, it was sometimes referred to as "the
calculus of infinitesimals", but that
usage is seldom seen today. Most basically, calculus is the study
of change, in the same way that geometry is the study of
space.

Calculus has widespread applications in science and engineering and is used to
solve problems for which algebra
alone is insufficient. Calculus builds on algebra, trigonometry, and analytic
geometry and includes two major branches, differential
calculus and integral
calculus, that are related by the
fundamental theorem of calculus. In more advanced mathematics,
calculus is usually called analysis
and is defined as the study of functions.

More generally, calculus (plural calculi) can
refer to any method or system of calculation guided by the symbolic
manipulation of expressions. Some examples of other well-known
calculi are propositional
calculus, predicate
calculus, relational
calculus, and lambda
calculus.

## History

### Development

The history of calculus falls into several
distinct time periods, most notably the ancient,
medieval, and
modern
periods. The ancient period introduced some of the ideas of
integral calculus, but does not seem to have developed these ideas
in a rigorous or systematic way. Calculating volumes and areas, the
basic function of integral calculus, can be traced back to the
Egyptian
Moscow papyrus (c. 1800 BC), in which an Egyptian successfully
calculated the volume of
a pyramidal frustum. From the school of
Greek
mathematics, Eudoxus
(c. 408−355 BC) used the method
of exhaustion, which prefigures the concept of the limit, to
calculate areas and volumes while Archimedes (c.
287−212 BC) developed this idea further, inventing
heuristics which
resemble integral calculus.
The method
of exhaustion was later used in China
by Liu
Hui in the 3rd century AD in order to find the area of a
circle. It was also used by Zu Chongzhi
in the 5th century AD, who used it to find the volume of a sphere. Around AD 1000, the
Islamic
mathematician Ibn
al-Haytham (Alhazen) was the first to derive the formula for
the sum of the fourth powers, and
using mathematical
induction, he developed a method that is readily generalizable
to finding the formula for the sum of any integral powers, which was
fundamental to the development of integral calculus. In the 12th
century, the Persian
mathematician
Sharaf
al-Din al-Tusi discovered the derivative of cubic
polynomials, an important result in differential calculus. In
the 14th century, Madhava
of Sangamagrama, along with other mathematician-astronomers of
the
Kerala school of astronomy and mathematics, described special
cases of Taylor
series, which are treated in the text Yuktibhasa.

In the modern period, independent discoveries in
calculus were being made in early 17th century Japan,
by mathematicians such as Seki Kowa, who
expanded upon the method
of exhaustion. In Europe, the second half of the 17th century
was a time of major innovation. Calculus provided a new opportunity
in mathematical
physics to solve long-standing problems. Several mathematicians
contributed to these breakthroughs, notably John Wallis
and Isaac
Barrow.
James Gregory proved a special case of the
second fundamental theorem of calculus in AD 1668.

Leibniz
and Newton
pulled these ideas together into a coherent whole and they are
usually credited with the independent and nearly simultaneous
invention of calculus. Newton was the first to apply calculus to
general physics and
Leibniz developed much of the notation used in calculus today; he
often spent days determining appropriate symbols for concepts. The
basic insight that both Newton and Leibniz had was the
fundamental theorem of calculus.

When Newton and Leibniz first published their
results, there was
great controversy over which mathematician (and therefore which
country) deserved credit. Newton derived his results first, but
Leibniz published first. Newton claimed Leibniz stole ideas from
his unpublished notes, which Newton had shared with a few members
of the Royal Society. This controversy divided English-speaking
mathematicians from continental mathematicians for many years, to
the detriment of English mathematics. A careful examination of the
papers of Leibniz and Newton shows that they arrived at their
results independently, with Leibniz starting first with integration
and Newton with differentiation. Today, both Newton and Leibniz are
given credit for developing calculus independently. It is Leibniz,
however, who gave the new discipline its name. Newton called his
calculus "the
science of fluxions".

Since the time of Leibniz and Newton, many
mathematicians have contributed to the continuing development of
calculus. In the 19th century, calculus was put on a much more
rigorous footing by mathematicians such as Cauchy, Riemann, and
Weierstrass. It
was also during this period that the ideas of calculus were
generalized to Euclidean
space and the complex
plane. Lebesgue further
generalized the notion of the integral.

Calculus is a ubiquitous topic in most modern
high schools and universities, and mathematicians around the world
continue to contribute to its development.

### Significance

While some of the ideas of calculus were
developed earlier, in Greece,
China,
India,
Iraq,
Persia, and Japan,
the modern use of calculus began in Europe, during the
17th century, when Isaac Newton
and Gottfried
Wilhelm Leibniz built on the work of earlier mathematicians to
introduce the basic principles of calculus. This work had a strong
impact on the development of physics.

Applications of differential calculus include
computations involving velocity and acceleration, the slope of a curve, and optimization.
Applications of integral calculus include computations involving
area, volume, arc length,
center of
mass, work, and
pressure. More advanced
applications include power series
and Fourier
series. Calculus can be used to compute the trajectory of a
shuttle docking at a space station or the amount of snow in a
driveway.

Calculus is also used to gain a more precise
understanding of the nature of space, time, and motion. For
centuries, mathematicians and philosophers wrestled with paradoxes
involving division
by zero or sums of infinitely many numbers. These questions
arise in the study of motion
and area. The ancient
Greek philosopher Zeno gave
several famous examples of such paradoxes.
Calculus provides tools, especially the limit
and the infinite
series, which resolve the paradoxes.

### Foundations

In mathematics, foundations refers to the rigorous development of a subject from precise axioms and definitions. Working out a rigorous foundation for calculus occupied mathematicians for much of the century following Newton and Leibniz and is still to some extent an active area of research today.There is more than one rigorous approach to the
foundation of calculus. The usual one is via the concept of
limits
defined on the continuum
of real
numbers. An alternative is nonstandard
analysis, in which the real number system is augmented with
infinitesimal and
infinite numbers. The
foundations of calculus are included in the field of real
analysis, which contains full definitions and proofs
of the theorems of calculus as well as generalizations such as
measure
theory and distribution
theory.

## Principles

### Limits and infinitesimals

Calculus is usually developed by manipulating very small quantities. Historically, the first method of doing so was by infinitesimals. These are objects which can be treated like numbers but which are, in some sense, "infinitely small". On a number line, these would be locations which are not zero, but which have zero distance from zero. No non-zero number is an infinitesimal, because its distance from zero is positive. Any multiple of an infinitesimal is still infinitely small, in other words, infinitesimals do not satisfy the Archimedean property. From this viewpoint, calculus is a collection of techniques for manipulating infinitesimals. This viewpoint fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. However, the concept was revived in the 20th century with the introduction of non-standard analysis and smooth infinitesimal analysis, which provided solid foundations for the manipulation of infinitesimals.In the 19th century, infinitesimals were replaced
by limits.
Limits describe the value of a function
at a certain input in terms of its values at nearby input. They
capture small-scale behavior, just like infinitesimals, but using
ordinary numbers. From this viewpoint, calculus is a collection of
techniques for manipulating certain limits. Infinitesimals get
replaced by very small numbers, and the infinitely small behavior
of the function is found by taking the limiting behavior for
smaller and smaller numbers. Limits are easy to put on rigorous
foundations, and for this reason they are the standard approach to
calculus.

### Derivatives

Differential calculus is the study of the definition, properties, and applications of the derivative or slope of a graph. The process of finding the derivative is called differentiation. In technical language, the derivative is a linear operator, which inputs a function and outputs a second function, so that at every point the value of the output is the slope of the input.The concept of the derivative is fundamentally
more advanced than the concepts encountered in algebra. In algebra,
students learn about functions which input a number and output
another number. For example, if the doubling function inputs 3,
then it outputs 6, while if the squaring function inputs 3, it
outputs 9. But the derivative inputs a function and outputs another
function. For example, if the derivative inputs the squaring
function, then it outputs the doubling function, because the
doubling function gives the slope of the squaring function at any
given point.

To understand the derivative, students must learn
mathematical notation. In mathematical notation, one common symbol
for the derivative of a function is an apostrophe-like mark called
prime.
Thus the derivative of f is f′ (spoken "f prime"). The
last sentence of the preceding paragraph, in mathematical notation,
would be written

\begin f(x) &= x^2 \\ f ' (x) &= 2x.
\end

If the input of a function is time, then the
derivative of that function is the rate at which the function
changes.

If a function is linear
(that is, if the graph
of the function is a straight line), then the function can be
written y = mx + b, where:

- m= \frac= = .

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 y divided by the change in x varies,
and we can use calculus to find an exact value at a given point.
(Note that y and f(x) represent the same thing: the output of the
function. This is known as function notation.) A line through two
points on a curve is called a secant line. The slope, or rise over
run, of a secant line can be expressed as

- m = = \,

where the coordinates of the first
point are (x, f(x)) and h is the horizontal distance between the
two points.

To determine the slope of the curve, we use the
limit:

- \lim_.

Working out one particular case, we find the
slope of the squaring function at the point where the input is 3
and the output is 9 (i.e., f(x) = x2, so f(3) = 9).

\begin f'(3)&=\lim_ \\ &=\lim_ \\
&=\lim_ \\ &=\lim_ (6 + h) \\ &= 6 \end

The slope of the squaring function at the point
(3, 9) is 6, that is to say, it is going up six times as fast as it
is going to the right.

The limit process just described can be
generalized to any point on the graph of any function. The
procedure can be visualized as in the following figure.

Here the function involved (drawn in red) is f(x)
= x3 − x. The tangent line (in green) which passes
through the point (−3/2, −15/8) has a slope of
23/4. Note that the vertical and horizontal scales in this image
are different.

### Integrals

Integral calculus is the study of the
definitions, properties, and applications of two related concepts,
the indefinite integral and the definite integral. The process of
finding the value of an integral is called integration. In
technical language, integral calculus studies two related linear
operators.

The indefinite integral is the antiderivative,
the inverse operation to the derivative. F is an indefinite
integral of f when f is a derivative of F. (This use of upper- and
lower-case letters for a function and its indefinite integral is
common in calculus.)

The definite integral inputs a function and
outputs a number, which gives the area between the graph of the
input and the x-axis. The
technical definition of the definite integral is the limit
of a sum of areas of rectangles, called a Riemann
sum.

A motivating example is the distances traveled in
a given time.

- \mathrm = \mathrm \cdot \mathrm

If the speed is constant, only multiplication is
needed, but if the speed changes, then we need a more powerful
method of finding the distance. One such method is to approximate
the distance traveled by breaking up the time into many short
intervals of time, then multiplying the time elapsed in each
interval by one of the speeds in that interval, and then taking the
sum (a Riemann sum)
of the approximate distance traveled in each interval. The basic
idea is that if only a short time elapses, then the speed will stay
more or less the same. However, a Riemann sum only gives an
approximation of the distance traveled. We must take the limit of
all such Riemann sums to find the exact distance traveled.

If f(x) in the diagram on the left represents
speed as it varies over time, the distance traveled between the
times represented by a and b is the area of the shaded region
s.

To approximate that area, an intuitive method
would be to divide up the distance between a and b into a number of
equal segments, the length of each segment represented by the
symbol Δx. For each small segment, we can choose one value of the
function f(x). Call that value h. Then the area of the rectangle
with base Δx and height h gives the distance (time Δx multiplied by
speed h) traveled in that segment. Associated with each segment is
the average value of the function above it, f(x)=h. The sum of all
such rectangles gives an approximation of the area between the axis
and the curve, which is an approximation of the total distance
traveled. A smaller value for Δx will give more rectangles and in
most cases a better approximation, but for an exact answer we need
to take a limit as Δx approaches zero.

The symbol of integration is \int \,, an
elongated S (which stands for "sum"). The definite integral is
written as:

- \int_a^b f(x)\, dx

and is read "the integral from a to b of f-of-x
with respect to x."

The indefinite integral, or antiderivative, is
written:

- \int f(x)\, dx.

Functions differing by only a constant have the
same derivative, and therefore the antiderivative of a given
function is actually a family of functions differing only by a
constant. Since the derivative of the function y = x² + C, where C
is any constant, is y′ = 2x, the antiderivative of the
latter is given by:

- \int 2x\, dx = x^2 + C.

### Fundamental theorem

The fundamental theorem of calculus states that differentiation and integration are inverse operations. More precisely, it relates the values of antiderivatives to definite integrals. Because it is usually easier to compute an antiderivative than to apply the definition of a definite integral, the Fundamental Theorem of Calculus provides a practical way of computing definite integrals. It can also be interpreted as a precise statement of the fact that differentiation is the inverse of integration.The Fundamental Theorem of Calculus states: If a
function f is continuous
on the interval [a, b] and if F is a function whose derivative is f
on the interval (a, b), then

- \int_^ f(x)\,dx = F(b) - F(a).

- \frac\int_a^x f(t)\, dt = f(x).

This realization, made by both Newton and
Leibniz,
who based their results on earlier work by Isaac
Barrow, was key to the massive proliferation of analytic
results after their work became known. The fundamental theorem
provides an algebraic method of computing many definite
integrals—without performing limit processes—by
finding formulas for antiderivatives. It is
also a prototype solution of a differential
equation. Differential equations relate an unknown function to
its derivatives, and are ubiquitous in the sciences.

## Applications

Calculus is used in every branch of the physical sciences, in computer science, statistics, engineering, economics, business, medicine, and in other fields wherever a problem can be mathematically modeled and an optimal solution is desired.Physics makes
particular use of calculus; all concepts in classical
mechanics are interrelated through calculus. The mass of an object of known density, the moment of
inertia of objects, as well as the total energy of an object
within a conservative field can be found by the use of calculus. In
the subfields of electricity and magnetism calculus can be used
to find the total flux of
electromagnetic fields. A more historical example of the use of
calculus in physics is Newton's
second law of motion, it expressly uses the term "rate of
change" which refers to the derivative: The rate of change of
momentum of a body is equal to the resultant force acting on the
body and is in the same direction. Even the common expression of
Newton's second law as
Force = Mass × Acceleration
involves differential calculus because acceleration can be
expressed as the derivative of velocity. Maxwell's theory of
electromagnetism and
Einstein's
theory of general
relativity are also expressed in the language of differential
calculus. Chemistry also uses calculus in determining reaction
rates and radioactive decay.

Calculus can be used in conjunction with other
mathematical disciplines. For example, it can be used with linear
algebra to find the "best fit" linear approximation for a set
of points in a domain.

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 analytic
geometry, the study of graphs of functions, calculus is used to
find high points and low points (maximums and minimums), slope,
concavity
and inflection
points.

In economics, calculus allows for the
determination of maximal profit by providing a way to easily
calculate both marginal
cost and marginal
revenue.

Calculus can be used to find approximate
solutions to equations, in methods such as Newton's
method, fixed
point iteration, and linear
approximation. For instance, spacecraft use a variation of the
Euler
method to approximate curved courses within zero gravity
environments.

## See also

sisterlinks Calculus### Lists

### Related topics

## References

### Notes

### Books

- Donald A. McQuarrie (2003). Mathematical Methods for Scientists and Engineers, University Science Books. ISBN 9781891389245
- James Stewart (2002). Calculus: Early Transcendentals, 5th ed., Brooks Cole. ISBN 9780534393212

## Other resources

### Further reading

- Courant, Richard ISBN 978-3540650584 Introduction to calculus and analysis 1.
- Robert A. Adams. (1999). ISBN 978-0-201-39607-2 Calculus: A complete course.
- Albers, Donald J.; Richard D. Anderson and Don O. Loftsgaarden, ed. (1986) Undergraduate Programs in the Mathematics and Computer Sciences: The 1985-1986 Survey, Mathematical Association of America No. 7.
- John L. Bell: A Primer of Infinitesimal Analysis, Cambridge University Press, 1998. ISBN 978-0-521-62401-5. Uses synthetic differential geometry and nilpotent infinitesimals.
- Florian Cajori, "The History of Notations of the Calculus." Annals of Mathematics, 2nd Ser., Vol. 25, No. 1 (Sep., 1923), pp. 1-46.
- Leonid P. Lebedev and Michael J. Cloud: "Approximating Perfection: a Mathematician's Journey into the World of Mechanics, Ch. 1: The Tools of Calculus", Princeton Univ. Press, 2004.
- Cliff Pickover. (2003). ISBN 978-0-471-26987-8 Calculus and Pizza: A Math Cookbook for the Hungry Mind.
- Michael Spivak. (September 1994). ISBN 978-0-914098-89-8 Calculus. Publish or Perish publishing.
- Silvanus P. Thompson and Martin Gardner. (1998). ISBN 978-0-312-18548-0 Calculus Made Easy.
- Mathematical Association of America. (1988). Calculus for a New Century; A Pump, Not a Filter, The Association, Stony Brook, NY. ED 300 252.
- Thomas/Finney. (1996). ISBN 978-0-201-53174-9 Calculus and Analytic geometry 9th, Addison Wesley.
- Weisstein, Eric W. "Second Fundamental Theorem of Calculus." From MathWorld--A Wolfram Web Resource.

### Online books

- Crowell, B. (2003). "Calculus" Light and Matter, Fullerton. Retrieved 6 May 2007 from http://www.lightandmatter.com/calc/calc.pdf
- Garrett, P. (2006). "Notes on first year calculus" University of Minnesota. Retrieved 6 May 2007 from http://www.math.umn.edu/~garrett/calculus/first_year/notes.pdf
- Faraz, H. (2006). "Understanding Calculus" Retrieved 6 May 2007 from Understanding Calculus, URL http://www.understandingcalculus.com/ (HTML only)
- Keisler, H. J. (2000). "Elementary Calculus: An Approach Using Infinitesimals" Retrieved 6 May 2007 from http://www.math.wisc.edu/~keisler/keislercalc1.pdf
- Mauch, S. (2004). "Sean's Applied Math Book" California Institute of Technology. Retrieved 6 May 2007 from http://www.cacr.caltech.edu/~sean/applied_math.pdf
- Sloughter, Dan (2000). "Difference Equations to Differential Equations: An introduction to calculus". Retrieved 6 May 2007 from http://math.furman.edu/~dcs/book/
- Stroyan, K.D. (2004). "A brief introduction to infinitesimal calculus" University of Iowa. Retrieved 6 May 2007 from http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm (HTML only)
- Strang, G. (1991). "Calculus" Massachusetts Institute of Technology. Retrieved 6 May 2007 from http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm.

### Web pages

- Calculus Made Easy (1914) by Silvanus P. ThompsonFull text in PDF
- The Online Calculus course for transfer, notes, video lectures, active forum at San Francisco State University by Professor Arek Goetz
- Calculus.org: The Calculus page at University of California, Davis — contains resources and links to other sites
- COW: Calculus on the Web at Temple University - contains resources ranging from pre-calculus and associated algebra
- Online Integrator (WebMathematica) from Wolfram Research
- The Role of Calculus in College Mathematics from ERICDigests.org
- OpenCourseWare Calculus from the Massachusetts Institute of Technology
- Infinitesimal Calculus — an article on its historical development, in Encyclopaedia of Mathematics, Michiel Hazewinkel ed. .

calculus in Afrikaans: Analise

calculus in Arabic: تفاضل

calculus in Min Nan: Bî-chek-hun

calculus in Bulgarian: Математически
анализ

calculus in Catalan: Càlcul infinitesimal

calculus in Danish: Infinitesimalregning

calculus in German: Infinitesimalrechnung

calculus in Modern Greek (1453-): Απειροστικός
λογισμός

calculus in Spanish: Cálculo

calculus in Esperanto: Infinitezima
kalkulo

calculus in Persian: حسابان

calculus in French: Calcul infinitésimal

calculus in Galician: Cálculo
infinitesimal

calculus in Korean: 미적분학

calculus in Hindi: कलन

calculus in Ido: Kalkulo

calculus in Indonesian: Kalkulus

calculus in Icelandic: Örsmæðareikningur

calculus in Italian: Calcolo
infinitesimale

calculus in Hebrew: חשבון אינפיניטסימלי

calculus in Javanese: Kalkulus

calculus in Latin: Calculus

calculus in Macedonian: Математичка
анализа

calculus in Malay (macrolanguage):
Kalkulus

calculus in Dutch: Analyse (wiskunde)

calculus in Norwegian: Matematisk analyse

calculus in Japanese: 微分積分学

calculus in Polish: Rachunek różniczkowy i
całkowy

calculus in Portuguese: Cálculo

calculus in Russian: Математический анализ

calculus in Scots: Calculus

calculus in Simple English: Calculus

calculus in Slovak: Infinitezimálny počet

calculus in Slovenian: Infinitezimalni
račun

calculus in Serbian: Калкулус

calculus in Finnish:
Differentiaalilaskenta

calculus in Swedish: Matematisk analys

calculus in Tamil: நுண்கணிதம்

calculus in Thai: แคลคูลัส

calculus in Ukrainian: Математичний аналіз

calculus in Vietnamese: Giải tích

calculus in Urdu: حسابان

calculus in Chinese:
微积分