A report on 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.

- Calculus
Archimedes used the method of exhaustion to calculate the area under a parabola.

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3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

Mathematics

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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, 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).

A definite integral of a function can be represented as the signed area of the region bounded by its graph.

Integral

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Integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data.

Integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data.

A definite integral of a function can be represented as the signed area of the region bounded by its graph.
Riemann–Darboux's integration (top) and Lebesgue integration (bottom)
A line integral sums together elements along a curve.
Numerical quadrature methods: rectangle method, trapezoidal rule, Romberg's method, Gaussian quadrature

Along with differentiation, integration is a fundamental, essential operation of calculus, and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others.

A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications in science and engineering.

Mathematical analysis

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

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.

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Infinitesimal

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

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.

Portrait by Christoph Bernhard Francke, 1695

Gottfried Wilhelm Leibniz

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

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.

An illustration of Desargues' theorem, a result in Euclidean and projective geometry

Geometry

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Geometry is, with arithmetic, one of the oldest branches of mathematics.

Geometry is, with arithmetic, one of the oldest branches of mathematics.

An illustration of Desargues' theorem, a result in Euclidean and projective geometry
A European and an Arab practicing geometry in the 15th century
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310).
An illustration of Euclid's parallel postulate
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.
A sphere is a surface that can be defined parametrically (by  or implicitly (by x2 + y2 + z2 − r2 = 0.)
Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Zhoubi Suanjing 500–200 BC. The Pythagorean theorem is a consequence of the Euclidean metric.
The Koch snowflake, with fractal dimension=log4/log3 and topological dimension=1
A tiling of the hyperbolic plane
Differential geometry uses tools from calculus to study problems involving curvature.
A thickening of the trefoil knot
Quintic Calabi–Yau threefold
Discrete geometry includes the study of various sphere packings.
The Cayley graph of the free group on two generators a and b
Bou Inania Madrasa, Fes, Morocco, zellige mosaic tiles forming elaborate geometric tessellations
The Pythagoreans discovered that the sides of a triangle could have incommensurable lengths.

This was a necessary precursor to the development of calculus and a precise quantitative science of physics.

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.

Derivative

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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).

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.

Portrait of Newton at 46 by Godfrey Kneller, 1689

Isaac Newton

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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.

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.

Archimedes Thoughtful
by Domenico Fetti (1620)

Archimedes

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Approximation of pi; defining and investigating the spiral that now bears his name; and devising a system using exponentiation for expressing very large numbers.

Approximation of pi; defining and investigating the spiral that now bears his name; and devising a system using exponentiation for expressing very large numbers.

Archimedes Thoughtful
by Domenico Fetti (1620)
The Death of Archimedes (1815) by Thomas Degeorge
Cicero Discovering the Tomb of Archimedes (1805) by Benjamin West
The Archimedes' screw can raise water efficiently.
Archimedes may have used mirrors acting collectively as a parabolic reflector to burn ships attacking Syracuse.
Archimedes calculates the side of the 12-gon from that of the hexagon and for each subsequent doubling of the sides of the regular polygon.
A proof that the area of the parabolic segment in the upper figure is equal to 4/3 that of the inscribed triangle in the lower figure from Quadrature of the Parabola.
Frontpage of Archimedes' Opera, in Greek and Latin, edited by David Rivault (1615).
A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases.
Ostomachion is a dissection puzzle found in the Archimedes Palimpsest.
In 1906, the Archimedes Palimpsest revealed works by Archimedes thought to have been lost.
The Fields Medal carries a portrait of Archimedes.
Artistic interpretation of Archimedes' mirror used to burn Roman ships.
Painting by Giulio Parigi, c. 1599.
Bronze statue of Archimedes in Berlin

Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including: the area of a circle; the surface area and volume of a sphere; area of an ellipse; the area under a parabola; the volume of a segment of a paraboloid of revolution; the volume of a segment of a hyperboloid of revolution; and the area of a spiral.

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.

Differential calculus

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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.