calculus

Topics in calculus
Fundamental theorem
Limits of functions
Continuity
Vector calculus
Tensor calculus
Mean value theorem
Differentiation
Product rule
Quotient rule
Chain rule
Implicit differentiation
Taylor's theorem
Related rates
Table of derivatives
Integration
Lists of integrals
Improper integrals
Integration by: parts, disks,
cylindrical shells, substitution,
trigonometric substitution
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", but that usage is seldom seen today. Calculus has widespread applications in science and engineering and is used to solve complicated 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.

History

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Sir Isaac Newton was one of the more famous inventors and contributors of calculus, with relation to his laws of motion and other mathematical physics concepts.

Development

Main article: History of calculus


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 worked out the volume of a pyramidal frustrum.[1] [2] Eudoxus (c. 408−355 BC) used the method of exhaustion, which prefigures the concept of the limit, to calculate areas and volumes. Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble integral calculus.[3] The method of exhaustion was rediscovered in China by Liu Hui in the 3rd century AD, who used it 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.[2]

In the early medieval period, the Indian mathematician Aryabhata used the notion of infinitesimals in AD 499 and expressed an astronomical problem in the form of a basic differential equation.[4] This equation eventually led Bhāskara II in the 12th century to develop an early derivative representing infinitesimal change, and he described an early form of "Rolle's theorem".[5] Around AD 1000, the Iraqi 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 for determining the general formula for the sum of any integral powers, which was fundamental to the development of integral calculus.[6] In the 12th century, the Persian mathematician Sharaf al-Din al-Tusi discovered the derivative of cubic polynomials, an important result in differential calculus.[7] 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,[8] which are treated in the text Yuktibhasa.[9][10][11]

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.

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Gottfried Wilhelm Leibniz was originally accused of plagiarism of Sir Isaac Newton's unpublished works, but is now regarded as an independent inventor and contributor towards calculus.


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 "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.[12]

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

Main article: Limit (mathematics)
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 is 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, 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

Main article: Derivative
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' (f prime). The last sentence of the preceding paragraph, in mathematical notation, would be written:

:.


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:

.


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 can use calculus to find an exact value at a given point. (Note that y and f(x) represent for the same thing: the output of the function.) 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:



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:



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. , so ).



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.

Integrals

Main article: Integral


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.



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.

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Integration can be thought of as measuring the area under a curve, defined by f(x), between two points (here a and b).


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 , 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)

.

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



Furthermore, 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.

Applications

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The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus
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.

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

Lists

Related topics

References

Notes

1. ^ His calculation was correct. We have no evidence how they found it; some, including Morris Kline (Mathematical thought from ancient to modern times Vol. I) suggest trial and error.
2. ^ Helmer Aslaksen. Why Calculus? National University of Singapore. See
3. ^ Archimedes, Method, in The Works of Archimedes ISBN 978-0-521-66160-7
4. ^ Aryabhata the Elder
5. ^ Ian G. Pearce. Bhaskaracharya II.
6. ^ Victor J. Katz (1995). "Ideas of Calculus in Islam and India", Mathematics Magazine 68 (3), pp. 163-174.
7. ^ J. L. Berggren (1990). "Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat", Journal of the American Oriental Society 110 (2), pp. 304-309.
8. ^ Madhava. Biography of Madhava. School of Mathematics and Statistics University of St Andrews, Scotland. Retrieved on 2006-09-13.
9. ^ An overview of Indian mathematics. Indian Maths. School of Mathematics and Statistics University of St Andrews, Scotland. Retrieved on 2006-07-07.
10. ^ Science and technology in free India. Government of Kerala — Kerala Call, September 2004. Prof.C.G.Ramachandran Nair. Retrieved on 2006-07-09.
11. ^ Charles Whish (1835). Transactions of the Royal Asiatic Society of Great Britain and Ireland. 
12. ^ UNESCO-World Data on Education [1]

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

  • 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.

Online books

Web pages

The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration.

The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite
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In mathematics, the limit of a function is a fundamental concept in analysis. Informally, a function f(x) has a limit L at a point p if the value of f(x) can be made as close to L as desired, by making x close enough to
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In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous.
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Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in a metric space with two or more dimensions (some results can only be applied to three dimensions[1]).
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The term tensor has slightly different meanings in mathematics and physics. In the mathematical fields of multilinear algebra and differential geometry, a tensor is a multilinear function.
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secant joining the endpoints of the interval [ab] is parallel to the tangent at c.]] In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative
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derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point.
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In calculus, the product rule also called Leibniz's law (see derivation), governs the differentiation of products of differentiable functions.

It may be stated thus:



or in the Leibniz notation thus:


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In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist.

If the function one wishes to differentiate, , can be written as


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In calculus, the chain rule is a formula for the derivative of the composite of two functions.

In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, then the rate of change of
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In mathematics, an implicit function is a generalization for the concept of a function in which the dependent variable may not be given explicitly in terms of the independent variable.
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Taylor's theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point.
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In differential calculus, related rates problems involve finding the rate at which a quantity is changing by relating that quantity to other quantities whose rates of change are known. The rate of change is usually with respect to time.
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The primary operation in differential calculus is finding a derivative. This table lists derivatives of many functions. In the following, f and g are differentiable functions, with respect to x, from the real numbers, and c is a real number.
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INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL) is detecting some of the most energetic radiation that comes from space. It is the most sensitive gamma ray observatory ever launched.
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See the following pages for lists of integrals:
  • List of integrals of rational functions
  • List of integrals of irrational functions
  • List of integrals of trigonometric functions
  • List of integrals of inverse trigonometric functions

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In calculus, an improper integral is the limit of a definite integral, as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits.
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In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, hopefully simpler, integrals. The rule arises from the product rule of differentiation.
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Disk integration is a means of calculating the volume of a solid of revolution, when integrating along the axis of revolution. This method models the generated 3 dimensional shape as a "stack" of an infinite number of disks (of varying radius) of infinitesimal thickness.
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Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution, when integrating along an axis perpendicular to the axis of revolution.

It makes use of the so-called "representative cylinder".
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In calculus, the substitution rule is a tool for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, the substitution rule is a relatively important tool for mathematicians.
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In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. One may use the trigonometric identities to simplify certain integrals containing the radical expressions:




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Latin}}} 
Official status
Official language of: Vatican City
Used for official purposes, but not spoken in everyday speech
Regulated by: Opus Fundatum Latinitas
Roman Catholic Church
Language codes
ISO 639-1: la
ISO 639-2: lat
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In mathematics, the concept of a "limit" is used to describe the behavior of a function as its argument either "gets close" to some point, or as it becomes arbitrarily large; or the behavior of a sequence's elements as their index increases indefinitely.
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derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point.
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INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL) is detecting some of the most energetic radiation that comes from space. It is the most sensitive gamma ray observatory ever launched.
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In mathematics, a series is often represented as the sum of a sequence of terms. That is, a series is represented as a list of numbers with addition operations between them, for example this arithmetic sequence:

1 + 2 + 3 + 4 + 5 + ... + 99 + 100.

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Science (from the Latin scientia, 'knowledge'), in the broadest sense, refers to any systematic knowledge or practice.[1] Examples of the broader use included political science and computer science, which are not incorrectly named, but rather named according to
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Engineering is the applied science of acquiring and applying knowledge to design, analysis, and/or construction of works for practical purposes. The American Engineers' Council for Professional Development, also known as ECPD,[1] (later ABET [2]
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Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic.
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