Calculus

Archimedes used the method of exhaustion to calculate the area under a parabola.
Alhazen, 11th-century Arab mathematician and physicist
Isaac Newton developed the use of calculus in his laws of motion and gravitation.
Gottfried Wilhelm Leibniz was the first to state clearly the rules of calculus.
Maria Gaetana Agnesi
The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus.

Mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

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Infinitesimal

Infinitesimal or infinitesimal number is a quantity that is closer to 0|zero than any standard real number, but that is not zero.

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Infinitesimals are a basic ingredient in calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity.

Function (mathematics)

Called the domain of the function and the set Y is called the codomain of the function.

Schematic depiction of a function described metaphorically as a "machine" or "black box" that for each input yields a corresponding output
The red curve is the graph of a function, because any vertical line has exactly one crossing point with the curve.
A function that associates any of the four colored shapes to its color.
The function mapping each year to its US motor vehicle death count, shown as a line chart
The same function, shown as a bar chart
Graph of a linear function
Graph of a polynomial function, here a quadratic function.
Graph of two trigonometric functions: sine and cosine.
Together, the two square roots of all nonnegative real numbers form a single smooth curve.
A composite function g(f(x)) can be visualized as the combination of two "machines".
A simple example of a function composition
Another composition. In this example, {{math|1=(g ∘ f )(c) = #}}.

Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity).

Isaac Newton

English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher") widely recognised as one of the greatest mathematicians and physicists of all time and among the most influential scientists.

Portrait of Newton at 46 by Godfrey Kneller, 1689
Sir Isaac Newton
Newton in 1702 by Godfrey Kneller
Replica of Newton's second reflecting telescope, which he presented to the Royal Society in 1672
Illustration of a dispersive prism separating white light into the colours of the spectrum, as discovered by Newton
Facsimile of a 1682 letter from Isaac Newton to Dr William Briggs, commenting on Briggs' A New Theory of Vision.
Engraving of a Portrait of Newton by John Vanderbank
Newton's own copy of his Principia, with hand-written corrections for the second edition, in the Wren Library at Trinity College, Cambridge.
Isaac Newton in old age in 1712, portrait by Sir James Thornhill
Coat of arms of the Newton family of Great Gonerby, Lincolnshire, afterwards used by Sir Isaac.
Newton's tomb monument in Westminster Abbey
A Wood engraving of Newton's famous steps under the apple tree.
Newton statue on display at the Oxford University Museum of Natural History
Newton (1795, detail) by William Blake. Newton is depicted critically as a "divine geometer".

Newton also made seminal contributions to optics, and shares credit with German mathematician Gottfried Wilhelm Leibniz for developing infinitesimal calculus.

Mathematical analysis

Branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.

A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications in science and engineering.
Archimedes used the method of exhaustion to compute the area inside a circle by finding the area of regular polygons with more and more sides. This was an early but informal example of a limit, one of the most basic concepts in mathematical analysis.

Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis.

Gottfried Wilhelm Leibniz

German polymath active as a mathematician, philosopher, scientist and diplomat.

Portrait by Christoph Bernhard Francke, 1695
Engraving of Gottfried Wilhelm Leibniz
Stepped reckoner
Leibniz's correspondence, papers and notes from 1669 to 1704, National Library of Poland.
A page from Leibniz's manuscript of the Monadology
A diagram of I Ching hexagrams sent to Leibniz from Joachim Bouvet. The Arabic numerals were added by Leibniz.
Leibnizstrasse street sign Berlin
Commercium philosophicum et mathematicum (1745), a collection of letters between Leibnitz and Johann Bernoulli

As a mathematician, his greatest achievement was the development of the main ideas of differential and integral calculus, independently of Isaac Newton's contemporaneous developments, and mathematicians have consistently favored Leibniz's notation as the conventional and more exact expression of calculus.

Mathematics

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
The distribution of prime numbers is a central point of study in number theory. This Ulam spiral serves to illustrate it, hinting, in particular, at the conditional independence between being prime and being a value of certain quadratic polynomials.
The quadratic formula expresses concisely the solutions of all quadratic equations
Rubik's cube: the study of its possible moves is a concrete application of group theory
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
A page from al-Khwārizmī's Algebra
Leonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World.
Leonhard Euler created and popularized much of the mathematical notation used today.
Carl Friedrich Gauss, known as the prince of mathematicians
The front side of the Fields Medal
Euler's identity, which American physicist Richard Feynman once called "the most remarkable formula in mathematics".

Mathematics is an area of knowledge that includes such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

Algebra

Quadratic formula.svg expresses the solution of the equation

The word algebra comes from the title of a book by Muhammad ibn Musa al-Khwarizmi.
A page from Al-Khwārizmī's al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala
Italian mathematician Girolamo Cardano published the solutions to the cubic and quartic equations in his 1545 book Ars magna.
Linear algebra lecture at the Aalto University
Algebraic expression notation:
 1 – power (exponent)
 2 – coefficient
 3 – term
 4 – operator
 5 – constant term
 x y c – variables/constants
The graph of a polynomial function of degree 3

Even though some methods, which had been developed much earlier, may be considered nowadays as algebra, the emergence of algebra and, soon thereafter, of infinitesimal calculus as subfields of mathematics only dates from the 16th or 17th century.

Area

Quantity that expresses the extent of a region on the plane or on a curved surface.

The combined area of these three shapes is approximately 15.57 squares.
This square and this disk both have the same area (see: squaring the circle).
A square metre quadrat made of PVC pipe.
Although there are 10 mm in 1 cm, there are 100 mm2 in 1 cm2.
The area of this rectangle is lw.
A diagram showing how a parallelogram can be re-arranged into the shape of a rectangle.
A parallelogram split into two equal triangles.
A circle can be divided into sectors which rearrange to form an approximate parallelogram.
Archimedes showed that the surface area of a sphere is exactly four times the area of a flat disk of the same radius, and the volume enclosed by the sphere is exactly 2/3 of the volume of a cylinder of the same height and radius.
Integration can be thought of as measuring the area under a curve, defined by f(x), between two points (here a and b).
The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions

For shapes with curved boundary, calculus is usually required to compute the area.

Differential calculus

The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line equals the derivative of the function at the marked point.

In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change.

Derivative

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

The graph of a function, drawn in black, and a tangent line to that graph, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point.
This function does not have a derivative at the marked point, as the function is not continuous there (specifically, it has a jump discontinuity).

Derivatives are a fundamental tool of calculus.