Applied mathematics

Efficient solutions to the vehicle routing problem require tools from combinatorial optimization and integer programming.
A numerical solution to the heat equation on a pump casing model using the finite element method.
Fluid mechanics is often considered a branch of applied mathematics and mechanical engineering.
Mathematical finance is concerned with the modelling of financial markets.
The Brown University Division of Applied Mathematics is the oldest applied math program in the U.S.
Applied mathematics has substantial overlap with statistics.

Application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry.

- Applied mathematics
Efficient solutions to the vehicle routing problem require tools from combinatorial optimization and integer programming.

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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

Area of knowledge that includes such topics as numbers , formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

Area of knowledge that includes such topics as numbers , formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

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".

Some areas of mathematics, such as statistics and game theory, are developed in close correlation with their applications and are often grouped under applied mathematics.

Pure mathematics studies the properties and structure of abstract objects, such as the E8 group, in group theory. This may be done without focusing on concrete applications of the concepts in the physical world.

Pure mathematics

Study of mathematical concepts independently of any application outside mathematics.

Study of mathematical concepts independently of any application outside mathematics.

Pure mathematics studies the properties and structure of abstract objects, such as the E8 group, in group theory. This may be done without focusing on concrete applications of the concepts in the physical world.
An illustration of the Banach–Tarski paradox, a famous result in pure mathematics. Although it is proven that it is possible to convert one sphere into two using nothing but cuts and rotations, the transformation involves objects that cannot exist in the physical world.

It follows that, presently, the distinction between pure and applied mathematics is more a philosophical point of view or a mathematician's preference rather than a rigid subdivision of mathematics.

Henri Poincaré
(photograph published in 1913)

Henri Poincaré

French mathematician, theoretical physicist, engineer, and philosopher of science.

French mathematician, theoretical physicist, engineer, and philosopher of science.

Henri Poincaré
(photograph published in 1913)
Plaque on the birthplace of Henri Poincaré at house number 117 on the Grande Rue in the city of Nancy
The young Henri Poincaré
The Poincaré family grave at the Cimetière du Montparnasse
Marie Curie and Poincaré talk at the 1911 Solvay Conference
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Photographic portrait of H. Poincaré by Henri Manuel
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As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics.

The InSight lander with solar panels deployed in a cleanroom

Engineering

Use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings.

Use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings.

The InSight lander with solar panels deployed in a cleanroom
The steam engine, the major driver in the Industrial Revolution, underscores the importance of engineering in modern history. This beam engine is on display in the Technical University of Madrid.
Relief map of the Citadel of Lille, designed in 1668 by Vauban, the foremost military engineer of his age.
The Ancient Romans built aqueducts to bring a steady supply of clean and fresh water to cities and towns in the empire.
A water-powered mine hoist used for raising ore, ca. 1556
The application of the steam engine allowed coke to be substituted for charcoal in iron making, lowering the cost of iron, which provided engineers with a new material for building bridges. This bridge was made of cast iron, which was soon displaced by less brittle wrought iron as a structural material
Jumbo Jet
Offshore platform, Gulf of Mexico
The solar furnace at Odeillo in the Pyrénées-Orientales in France can reach temperatures up to 3500 C
Hoover Dam
Electric motor
Design of a turbine requires collaboration of engineers from many fields, as the system involves mechanical, electro-magnetic and chemical processes. The blades, rotor and stator as well as the steam cycle all need to be carefully designed and optimized.
A drawing for a booster engine for steam locomotives. Engineering is applied to design, with emphasis on function and the utilization of mathematics and science.
A computer simulation of high velocity air flow around a Space Shuttle orbiter during re-entry. Solutions to the flow require modelling of the combined effects of fluid flow and the heat equations.
Graphic representation of a minute fraction of the WWW, demonstrating hyperlinks
Robotic Kismet can produce a range of facial expressions.
Radar, GPS, lidar, ... are all combined to provide proper navigation and obstacle avoidance (vehicle developed for 2007 DARPA Urban Challenge)
Engineers, scientists and technicians at work on target positioner inside National Ignition Facility (NIF) target chamber
The International Space Station is used to conduct science experiments in space
A 3 tesla clinical MRI scanner.
Genetically engineered mice expressing green fluorescent protein, which glows green under blue light. The central mouse is wild-type.
Leonardo da Vinci, seen here in a self-portrait, has been described as the epitome of the artist/engineer. He is also known for his studies on human anatomy and physiology.

The discipline of engineering encompasses a broad range of more specialized fields of engineering, each with a more specific emphasis on particular areas of applied mathematics, applied science, and types of application.

Graph of a 
given by z = f(x, y) = −(x² + y²) + 4. The global maximum at (x, y, z) = (0, 0, 4) is indicated by a blue dot.

Mathematical optimization

Selection of a best element, with regard to some criterion, from some set of available alternatives.

Selection of a best element, with regard to some criterion, from some set of available alternatives.

Graph of a 
given by z = f(x, y) = −(x² + y²) + 4. The global maximum at (x, y, z) = (0, 0, 4) is indicated by a blue dot.
Nelder-Mead minimum search of Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lowest ( best) value.

The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics.

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

Asymptotic analysis

Method of describing limiting behavior.

Method of describing limiting behavior.

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

In applied mathematics, asymptotic analysis is used to build numerical methods to approximate equation solutions.

Coat of arms

Brown University

Private Ivy League research university in Providence, Rhode Island.

Private Ivy League research university in Providence, Rhode Island.

Coat of arms
This 1792 engraving is the first published image of Brown. University Hall stands on right while the President's House sits on the left.
Following the gift of Nicholas Brown, Jr. (Class of 1786), the university was renamed in his honor
Slavery Memorial was designed by Martin Puryear and dedicated in 2014
The Van Wickle Gates stand at the crest of College Hill
The John Hay Library is home to rare books, special collections, and the university archives
The John Carter Brown Library is one of the world's leading repositories of books, maps, and manuscripts relating to the colonial Americas
The galleries of Brown's anthropology museum, the Haffenreffer, are located in Manning Hall
Three dormitories, Metcalf Hall (1919), Andrews Hall (1947), and Miller Hall (1910), formed the heart of Pembroke College and now serve as freshman residences
Robinson Hall (1878) was designed by Walker and Gould in the Venetian Gothic style to house Brown's library
The List Art Center, built 1969–71, designed by Philip Johnson, houses Brown's Department of Visual Art and the David Winton Bell Gallery
The Granoff Center, designed by Diller Scofidio + Renfro
Lyman Hall, built 1890–92, houses the Department of Theatre Arts and Performance Studies
Aerial view of the Brown University English department
The Brown Computing Laboratory, designed by Philip Johnson
The Brown University Engineering Research Center, completed in 2018 and designed by KieranTimberlake
Pembroke Hall (1897) houses the administrative offices of the Pembroke Center for Teaching and Research on Women
Sayles Hall on the Main Green
The Alpert Medical School building on Richmond Street
The primary building of the Brown University School of Public Health viewed from across the Providence River
Many Spring Weekend events are hosted on Brown's Main Green
Ladd Observatory, built 1890–1891, is used by Brown Space Engineering, a student group focused on Aerospace engineering
The Brown University Band was founded in 1924
The Sarah Doyle Women's Center
The 1879 Brown baseball varsity, with W.E. White seated second from right. White's appearance in an 1879 major league game may be the first person of color to play professional baseball, 68 years before Jackie Robinson
Horace Mann, class of 1819, regarded as the father of American public education
Samuel Gridley Howe, class of 1821, abolitionist and advocate for the blind
John Hay, class of 1858, private secretary to Abraham Lincoln and U.S. Secretary of State
Charles Evans Hughes, class of 1881, Chief Justice of the United States and U.S. Secretary of State
John D. Rockefeller Jr., class of 1897, philanthropist and developer of Rockefeller Center
Lois Lowry, class of 1958, Newbery Medal-winning author of The Giver and Number the Stars
Ted Turner, class of 1960, founder of CNN, TBS, and WCW and philanthropist
John Sculley, class of 1961, former CEO of Apple Inc. and president of PepsiCo
Janet Yellen, class of 1967, first woman to serve as Chair of the Federal Reserve and U.S. Secretary of the Treasury
André Leon Talley, class of 1972, former editor-at-large and creative director of Vogue
Brian Moynihan, class of 1981, chairman and CEO of Bank of America
Ira Glass, class of 1982, radio personality and host of This American Life
Jim Yong Kim, class of 1982, 12th Pres. of the World Bank, 17th Pres. of Dartmouth
Dara Khosrowshahi, class of 1991, CEO of Uber, former CEO of Expedia Group
John F. Kennedy Jr., class of 1983, lawyer, journalist, and magazine publisher
Davis Guggenheim, class of 1986, Oscar-winning documentary filmmaker
Laura Linney, class of 1986, actress, recipient of 4 Emmy Awards and 3 time Oscar nominee
Julie Bowen, class of 1991, actress, six time Emmy Award nominee
Tracee Ellis Ross, class of 1994, actress, model, comedienne, and television host
Andrew Yang, class of 1996, businessman and U.S. presidential candidate
Chris Hayes, class of 2001, political commentator and host of All In with Chris Hayes
John Krasinski, class of 2001, actor, director, producer, and screenwriter
A. G. Sulzberger, class of 2003, publisher of The New York Times
Emma Watson, class of 2014, actress, model, activist

The university is home to the oldest applied mathematics program in the United States, the oldest engineering program in the Ivy League, and the third-oldest medical program in New England.

At NASA's StarChild Learning Center, c. 1980s

Stephen Hawking

English theoretical physicist, cosmologist, and author who, at the time of his death, was director of research at the Centre for Theoretical Cosmology at the University of Cambridge.

English theoretical physicist, cosmologist, and author who, at the time of his death, was director of research at the Centre for Theoretical Cosmology at the University of Cambridge.

At NASA's StarChild Learning Center, c. 1980s
Stephen Hawking in 1966
Hawking at an ALS convention in San Francisco in the 1980s
Hawking with string theorists David Gross and Edward Witten at the Strings Conference in January 2001, TIFR, India
Hawking at the Bibliothèque nationale de France to inaugurate the Laboratory of Astronomy and Particles in Paris, and the French release of his work God Created the Integers, 5 May 2006
Hawking with University of Oxford librarian Richard Ovenden (left) and naturalist David Attenborough (right) at the opening of the Weston Library, Oxford, in March 2015. Ovenden awarded the Bodley Medal to Hawking and Attenborough at the ceremony.
Hawking holding a public lecture at the Stockholm Waterfront congress centre, 24 August 2015
Hawking taking a zero-gravity flight in a reduced-gravity aircraft, April 2007
Stephen Hawking's memorial stone in Westminster Abbey
President Barack Obama talks with Hawking in the White House before a ceremony presenting him with the Presidential Medal of Freedom on 12 August 2009
Hawking in Monty Python's “Galaxy Song” video at the comedy troupe's 2014 reunion show, Monty Python Live (Mostly)
Hawking being presented by his daughter Lucy Hawking at the lecture he gave for NASA's 50th anniversary, 2008
The Blackboard from Hawking's Office on display in the Science Museum

In October 1962, he began his graduate work at Trinity Hall, Cambridge, where in March 1966, he obtained his PhD degree in applied mathematics and theoretical physics, specialising in general relativity and cosmology.

Francisco de Goya, The Sleep of Reason Produces Monsters (El sueño de la razón produce monstruos), c. 1797

Statistician

Person who works with theoretical or applied statistics.

Person who works with theoretical or applied statistics.

Francisco de Goya, The Sleep of Reason Produces Monsters (El sueño de la razón produce monstruos), c. 1797

Additionally, there is a substantial number of people who use statistics and data analysis in their work but have job titles other than statistician, such as actuaries, applied mathematicians, economists, data scientists, data analysts (predictive analytics), financial analysts, psychometricians, sociologists, epidemiologists, and quantitative psychologists.

A black and white rendition of the Yale Babylonian Collection's Tablet YBC 7289 (c. 1800–1600 BCE), showing a Babylonian approximation to the square root of 2 (1 24 51 10 w: sexagesimal) in the context of Pythagoras' Theorem for an isosceles triangle. The tablet also gives an example where one side of the square is 30, and the resulting diagonal is 42 25 35 or 42.4263888.

Computational mathematics

Computational mathematics involves mathematical research in mathematics as well as in areas of science where computation plays a central and essential role, and emphasizes algorithms, numerical methods, and symbolic computations.

Computational mathematics involves mathematical research in mathematics as well as in areas of science where computation plays a central and essential role, and emphasizes algorithms, numerical methods, and symbolic computations.

A black and white rendition of the Yale Babylonian Collection's Tablet YBC 7289 (c. 1800–1600 BCE), showing a Babylonian approximation to the square root of 2 (1 24 51 10 w: sexagesimal) in the context of Pythagoras' Theorem for an isosceles triangle. The tablet also gives an example where one side of the square is 30, and the resulting diagonal is 42 25 35 or 42.4263888.

Computational applied mathematics consists roughly of using mathematics for allowing and improving computer computation in applied mathematics.