Calculus

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Calculus [from Latin, literally "pebble" (used in reckoning)] is a major area in mathematics, with applications in science, engineering, business, and medicine. It builds on analytic geometry, and extends that field by introducing the concept of the limit, which allows control over arbitrarily small and arbitrarily large numbers. Calculus includes two major branches, differential calculus and integral calculus, related by the Fundamental Theorem of Calculus.

Applications of differential calculus include computations involving position, velocity, and acceleration, the slope of a curve, related rates, and optimization. Applications of integral calculus include computations involving area, volume, arc length, center of mass, work, and force. 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.

While some of the ideas of calculus were developed earlier, in ancient Greece and in India, 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.

[edit] Differentiation

Main article: Derivative

Differential calculus defines a linear operator called the derivative, and describes its properties and applications.

Tangent line at (x, f(x)). The derivative f ' (x) of a curve at a point is the slope (rise over run) of the line tangent to that curve at that point.

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 functions gives the slope of the squaring function at any given point.

To understand the derivative, students must master 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 ' (f prime). The last sentence of the preceding paragraph, in mathematical notation, would be written:

If f(x) = x², then f ' (x) = 2x.

The derivative of a function measures the slope of the function. If the input of the function is time, then the derivative measures 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 f(x) = m x + b, where:

.

This gives an exact value for the slope of a straight line. If the function is not a straight line, however then the change in y divided by the change in x varies, and we need calculus to find an exact value at a given point.

The slope, or rise over run, can be expressed as:

To determine the instantaneous rate of change we use the limit:

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.

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.

For more technical information on the limit and the derivative, see their respective articles.

[edit] Integration

Main article: Integral

Integration is applied in two (related) ways.

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

The definite integral is the limit of a sum, called a Riemann sum. A motivating example is the distances traveled in a given time.

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 (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.

Integration can be thought of as measuring the area under a curve, defined by f(x), between two points (here a and b) by subdividing the area into ever-smaller slices and then adding them all up.

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 in to 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 goes to zero.

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

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

The indefinite integral, or antiderivative, is written:

.

Since the derivative of the function y = x² + C is y ' = 2x (where C is any constant)

.

For a more technical discussion, see the main article, integral.

[edit] Fundamental theorem

Main article: Fundamental theorem of calculus

The fundamental theorem of calculus states that differentiation and integration are, in a certain sense, inverse operations. More precisely, if one defines one function as the integral of another continuous function, then differentiating the newly defined function returns the function you started with. Furthermore, if you want to find the value of a definite integral, you usually do so by evaluating an antiderivative.

Here is the mathematical formulation of the Fundamental Theorem of Calculus: 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

Also, for every x in the interval [a, b],

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.

[edit] 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 generalisations such as measure theory and distribution theory.

[edit] History

Main article: History of calculus

The history of calculus falls into several distinct periods, most notably the Greek, Indian, and European periods.

The Greek period introduced some of the ideas of integral calculus, but does not seem to have developed these ideas in a rigorous or systematic way. Eudoxus (circa 408 BCE - circa 355 BCE) used the method of exhaustion, which prefigures the concept of the limit, to calculate areas and volumes. Archimedes (circa 287 BCE - 212 BCE) developed this idea further, inventing heuristics which resemble integral calculus.^[1]

Indian mathematics, largely unknown in the west until the 20th century, produced a number of works with some ideas of calculus. The mathematician-astronomer Aryabhata in 499 CE used a notion of infinitesimals and expressed an astronomical problem in the form of a basic differential equation.^[2] Manjula, in the 10th century, elaborated on this differential equation in a commentary. This equation eventually led Bhaskara in the 12th century to develop a proto-derivative representing infinitesimal change, and describe an early form of "Rolle's theorem".^[3] In the 14th century, Madhava, along with other mathematician-astronomers of the Kerala School, described special cases of Taylor series^[4] which are treated in the text Yuktibhasa.^[5][6][7]

Sir Isaac Newton

Gottfried Wilhelm Leibniz

The second half of the 17th century was a time of major innovation in Europe. 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 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 fundamental 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. 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 "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.

There were isolated instances mathematicians in other cultures who made independent discoveries in calculus. For example, in Japan Seki Kowa (circa 1637 - 1708) expanded further upon Eudoxus's method of exhaustion.

Today, calculus is studied around the world, and mathematicians worldwide contribute to new advances in calculus.

[edit] Applications

Please help improve this article by expanding this section. Further information might be found on the talk page or at requests for expansion. Please remove this message once the section has been expanded. This article has been tagged since March 2007.

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. Newton's second law of motion expressly uses the term "rate of change" which is 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 is 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.

[edit] See also

[edit] Lists

• List of basic calculus topics
• List of basic calculus equations and formulas
• List of calculus topics
• Publications in calculus

[edit] Related topics

• Calculus with polynomials
• Differential geometry
• Mathematics
• Multivariable calculus
• Non-standard analysis
• Precalculus (mathematical education)

[edit] References

• 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

[edit] Further reading

• 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. (Sept 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.

[edit] External links

[edit] Online books

Wikibooks has more on the topic of

Calculus

Wikisource has an original article from the 1911 Encyclopædia Britannica about:

Infinitesimal Calculus

• Crowell, Benjamin, Calculus, Fullerton College, an online textbook
• Garrett, Paul, Notes on First-Year Calculus
• Hussain, Faraz, Understanding Calculus, a complete online book with a conceptual focus
• Keisler, H. Jerome, Elementary Calculus: An Approach Using Infinitesimals, University of Wisconsin
• Mauch, Sean, Sean's Applied Math Book, CIT, an online textbook that includes a complete introduction to calculus
• Sloughter, Dan, Difference Equations to Differential Equations, an introduction to calculus
• Stroyan, K.D., A Brief Introduction to Infinitesimal Calculus, University of Iowa
• Gilbert Strang, A useful textbook resource for educators and self-learners alike, MIT OpenCourseWare, there is also an online Instructor's Manual and a student Study Guide included.

[edit] Web pages

• calculus.org: The calculus page at University of California, Davis — contains resources and links to other sites
• Eric W. Weisstein, Calculus at MathWorld.
• COW: Calculus on the Web
• Topics on Calculus at PlanetMath.
• Online Integrator by Mathematica
• The Role of Calculus in College Mathematics
• Calculus on MIT OpenCourseWare
• Infinitesimal Calculus — an article on its historical development, in Encyclopaedia of Mathematics, Michiel Hazewinkel ed. .