FACTOID # 30: If Alaska were its own country, it would be the 26th largest in total area, slightly larger than Iran.

 Home Encyclopedia Statistics States A-Z Flags Maps FAQ About

 WHAT'S NEW RELATED ARTICLES People who viewed "Calculus" also viewed:

SEARCH ALL

Search encyclopedia, statistics and forums:

(* = Graphable)

Encyclopedia > Calculus

More generally, calculus can refer to any method or system of calculation.

## History

Aryabhatta along with other Indian mathematicians over the centuries made important contribution to the field of calculus.
Sir Isaac Newton is one of the most famous contributors to the development of calculus, with, among other things, the use of calculus in his laws of motion and gravitation.

Image File history File links Download high-resolution version (951x1361, 193 KB)  Summary This image of a public statue in IUCAA Pune was photographed in May 2006 by myself, and I release all rights. ... Image File history File links Download high-resolution version (951x1361, 193 KB)  Summary This image of a public statue in IUCAA Pune was photographed in May 2006 by myself, and I release all rights. ... Aryabhata (आर्यभट) (Āryabhaṭa) is the first of the great astronomers of the classical age of India. ... Godfrey Knellers portrait of Isaac Newton (1689) oil on canvas. ... Godfrey Knellers portrait of Isaac Newton (1689) oil on canvas. ... Sir Isaac Newton in Knellers portrait of 1689. ...

### Development

Main article: History of calculus

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. Kowa Seki (Seki Takakazu, 関 孝和) (1642? – October 24, 1708) was a Japanese mathematician. ... The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. ... Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories. ... John Wallis John Wallis (November 22, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the development of modern calculus. ... Isaac Barrow (October 1630 - May 4, 1677) was an English divine, scholar and mathematician who is generally given minor credit for his role in the development of modern calculus; in particular, for his work regarding the tangent; for example, Barrow is given credit for being the first to calculate the... James Gregory For other people with the same name, see James Gregory. ... The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ...

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. Image File history File links Gottfried_Wilhelm_von_Leibniz. ... Image File history File links Gottfried_Wilhelm_von_Leibniz. ... Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... For other uses, see Plagiarism (disambiguation). ... Leibniz redirects here. ... Sir Isaac Newton FRS (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ...

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". Isaac Newton began working on a form of the calculus in 1666. ... Method of Fluxions was a book by Isaac Newton. ...

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. Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... Bernhard Riemann. ... Karl Theodor Wilhelm Weierstraß (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. (The letter ß may be transliterated as ss; one often writes Weierstrass. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... Henri Léon Lebesgue (June 28, 1875 - July 26, 1941) was a French mathematician, most famous for his theory of integration. ...

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. In the history of mathematics, Islamic mathematics or Arabic mathematics refers to the mathematics developed by the Islamic civilization between 622 and 1600. ... For other uses, see Europe (disambiguation). ... Sir Isaac Newton FRS (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ...

### 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. For the medical term see rigor (medicine) Rigour (American English: rigor) has a number of meanings in relation to intellectual life and discourse. ...

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. Wikibooks Calculus has a page on the topic of Limits 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 sequences elements as... In mathematics, the word continuum sometimes denotes the real line. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In the most restricted sense, nonstandard analysis or non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ...

## 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 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. Wikibooks Calculus has a page on the topic of Limits 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 sequences elements as... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... In mathematics (particularly abstract algebra), the Archimedean property is a property held by some ordered algebraic structures, and in particular by the ordered field of real numbers. ... Non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a natural... Smooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms 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. Wikibooks Calculus has a page on the topic of Limits 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 sequences elements as... This article is about functions in mathematics. ...

### Derivatives

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.
Main article: Derivative

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 This article is not about the symbol for the set of prime numbers, â„™. The prime (â€², Unicode U+2032, &prime;) is a symbol with many mathematical uses: A complement in set theory: Aâ€² is the complement of the set A A point related to another (e. ...

begin{align} f(x) &= x^2 f ' (x) &= 2x. end{align}

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: A linear function is a mathematical function term of the form: f(x) = m x + c where c is a constant. ...

$m= frac{mbox{rise}}{mbox{run}}= {mbox{change in } y over mbox{change in } x} = {Delta y over{Delta x}}$.

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

$m = {f(x+h) - f(x)over{(x+h) - x}} = {f(x+h) - f(x)over{h}},$

where the coordinates of the first point are (x, f(x)) and h is the horizontal distance between the two points. See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...

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

$lim_{h to 0}{f(x+h) - f(x)over{h}}$.

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{align} f'(3)&=lim_{h to 0}{(3+h)^2 - 9over{h}} &=lim_{h to 0}{9 + 6h + h^2 - 9over{h}} &=lim_{h to 0}{6h + h^2over{h}} &=lim_{h to 0} (6 + h) &= 6 end{align}

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.

Tangent line as a limit of secant lines. The derivative f′(x) of a curve at a point is the slope of the line tangent to that curve at that point. This slope is determined by considering the limiting value of the slopes of secant lines.

Here the function involved (drawn in red) is f(x) = x3x. 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

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. This article is about the concept of integrals in calculus. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

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. Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... Wikibooks Calculus has a page on the topic of Limits 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 sequences elements as... In mathematics, a Riemann sum is a method for approximating the values of integrals. ...

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

$mathrm{Distance} = mathrm{Speed} cdot mathrm{Time}$

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. In mathematics, a Riemann sum is a method for approximating the values of integrals. ...

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. Image File history File links Integral_as_region_under_curve. ... Image File history File links Integral_as_region_under_curve. ...

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:

$int 2x, dx = x^2 + C$.

An undetermined constant like C in the antiderivative is known as a constant of integration. In calculus, the indefinite integral of a given function (i. ...

### 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 specifies the relationship between the two central operations of calculus, differentiation and integration. ... The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and 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 In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

$int_{a}^{b} f(x),dx = F(b) - F(a).$

Furthermore, for every x in the interval (a, b),

$frac{d}{dx}int_a^x f(t), dt = f(x).$

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. Sir Isaac Newton FRS (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Leibniz redirects here. ... Isaac Barrow (October 1630 - May 4, 1677) was an English divine, scholar and mathematician who is generally given minor credit for his role in the development of modern calculus; in particular, for his work regarding the tangent; for example, Barrow is given credit for being the first to calculate the... In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ... Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ...

## Applications

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

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. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... For other uses, see Mass (disambiguation). ... For other uses, see Density (disambiguation). ... Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg m2, Former British units slug ft2), is the rotational analog of mass. ... Electricity (from New Latin Ä“lectricus, amberlike) is a general term for a variety of phenomena resulting from the presence and flow of electric charge. ... For other senses of this word, see magnetism (disambiguation). ... flux in science and mathematics. ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... â€œEinsteinâ€ redirects here. ... For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. ...

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. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...

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. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ... In calculus, a differentiable function f is convex on an interval if its derivative function f â€² is increasing on that interval: a convex function has an increasing slope. ... Plot of y = x3 with inflection point of (0,0). ...

In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both marginal cost and marginal revenue. In economics and finance, marginal cost is the change in total cost that arises when the quantity produced changes by one unit. ... In microeconomics, Marginal Revenue (MR) is the extra revenue that an additional unit of product will bring a firm. ...

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. In numerical analysis, Newtons method (also known as the Newtonâ€“Raphson method or the Newtonâ€“Fourier method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ... In numerical analysis, fixed point iteration is a method of computing fixed points of functions. ... Linear approximation is a method of approximating otherwise difficult to find values of a mathematical function by taking the value on a nearby tangent line instead of the function itself. ... In mathematics and computational science, Euler integration is the most basic kind of numerical integration for calculating trajectories from forces at discrete timesteps. ...

Find more about Calculus on Wikipedia's sister projects:
Dictionary definitions
Textbooks
Quotations
Source texts
Images and media
News stories
Learning resources

Wikipedia does not have an article with this exact name. ... Image File history File links Wikibooks-logo. ... Image File history File links Wikiquote-logo. ... Image File history File links Wikisource-logo. ... Image File history File links Commons-logo. ... Image File history File links WikiNews-Logo. ... Image File history File links Wikiversity-logo-Snorky. ...

### Lists

For a more comprehensive list, see the List of calculus topics. ... The primary operation in differential calculus is finding a derivative. ... This is a list of calculus topics. ... This is a list of important publications in mathematics, organized by field. ... It has been suggested that this article or section be merged with List of integrals. ...

### Related topics

In mathematics, polynomials are perhaps the simplest functions with which to do calculus. ... It has been suggested that this article be split into multiple articles. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Multivariable calculus is the extension of calculus in one variable to calculus in several variables: the functions which are differentiated and integrated involve several variables rather than one variable. ... Non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a natural... In mathematics education, precalculus, an advanced form of secondary school algebra, is a foundational mathematical discipline. ... Mathematics education is a term that refers both to the practice of teaching and learning mathematics, as well as to a field of scholarly research on this practice. ... Product integrals are a multiplicative version of standard integrals of infinitesimal calculus. ... Stochastic calculus is a branch of mathematics that operates on stochastic processes. ...

## References

### Notes

1. ^ There is no exact evidence on how it was done; some, including Morris Kline (Mathematical thought from ancient to modern times Vol. I) suggest trial and error.
2. ^ a b 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]

Morris Kline (1 May 1908 â€“ 10 June 1992) was a Professor of Mathematics, a writer on the history, philosophy, and teaching of mathematics, and also a popularizer of mathematical subjects. ... Malay name Malay: Universiti Nasional Singapura Tamil name Tamil: à®šà®¿à®™à¯à®•à®ªà¯à®ªà¯‚à®°à¯ à®¤à¯‡à®šà®¿à®¯ à®ªà®²à¯à®•à®²à¯ˆà®•à¯à®•à®´à®•à®®à¯ University Cultural Centre The National University of Singapore (Abbreviation: NUS) is Singapores oldest university. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 256th day of the year (257th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 188th day of the year (189th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 190th day of the year (191st in leap years) in the Gregorian calendar. ... UNESCO (United Nations Educational, Scientific and Cultural Organization) is a specialized agency of the United Nations established in 1945. ...

### 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

• 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.

Richard Courant (born January 8, 1888 at Lublinitz, today Poland, died January 27, 1972 at New York/USA) was a German and American mathematician. ... Smooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of infinitesimals. ... Florian Cajori at Colorado College Florian Cajori was born February 28, 1859 in St Aignan (near Thusis), Graubünden, Switzerland. ... Clifford A. Pickover is a writer in the fields of science, mathematics, and science fiction. ... Michael David Spivak is a mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press. ... Silvanus Phillips Thompson (June 19, 1851 â€“ June 12, 1916). ... Martin Gardner (b. ... The Mathematical Association of America (MAA) is a professional society that focuses on undergraduate mathematics education. ...

### Online books

is the 126th day of the year (127th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 126th day of the year (127th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 126th day of the year (127th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 126th day of the year (127th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 126th day of the year (127th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 126th day of the year (127th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 126th day of the year (127th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 126th day of the year (127th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ...

### Web pages

Results from FactBites:

 calculus.org - THE CALCULUS PAGE . (1238 words) Calculus demonstrations that you can use in the classroom to spice up your lectures and prevent those nodding heads. Single Variable Calculus Mika Seppälä of Florida State University and the University of Helsinki presents classroom type notes on calculus, in pdf and powerpoint format. Math Applets for Calculus at SLU: Mike May at St. Louis University has a selection of applets to illustrate important concepts of single and multivariable calculus.
 Calculus graphics -- Douglas N. Arnold (1444 words) The diagram illustrates the local accuracy of the tangent line approximation to a smooth curve, or--otherwise stated--the closeness of the differential of a function to the difference of function values due to a small increment of the independent variable. The proof is based on a diagram depicting a circular sector in the unit circle together with an inscribed and a circumscribed triangle. A brief graphical exploration of a continuous, nowhere differentiable function fits very well in the first semester of calculus, for example, to provide a strong counterexample to the converse of the theorem that differentiability implies continuity; or to show that it is only differentiable functions which look like straight lines under the microscope.
More results at FactBites »

Share your thoughts, questions and commentary here