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Mathematics is the study of numbers, quantity, space, pattern, structure, and change. Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.

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Diophantus-II-8-Fermat.jpg
Problem II.8 in the Arithmetica by Diophantus, annotated with Fermat's comment, which became Fermat's Last Theorem
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Fermat's Last Theorem is one of the most famous theorems in the history of mathematics. It states that:

has no solutions in non-zero integers , , and when is an integer greater than 2.

Despite how closely the problem is related to the Pythagorean theorem, which has infinite solutions and hundreds of proofs, Fermat's subtle variation is much more difficult to prove. Still, the problem itself is easily understood even by schoolchildren, making it all the more frustrating and generating perhaps more incorrect proofs than any other problem in the history of mathematics.

The 17th-century mathematician Pierre de Fermat wrote in 1637 in his copy of Bachet's translation of the famous Arithmetica of Diophantus: "I have a truly marvelous proof of this proposition which this margin is too narrow to contain." However, no correct proof was found for 357 years, until it was finally proven using very deep methods by Andrew Wiles in 1995 (after a failed attempt a year before).

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animation illustrating the meaning of a line integral of a two-dimensional scalar field

A line integral is an integral where the function to be integrated, be it a scalar field as here or a vector field, is evaluated along a curve. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). A detailed explanation of the animation is available. The key insight is that line integrals may be reduced to simpler definite integrals. (See also a similar animation illustrating a line integral of a vector field.) Many formulas in elementary physics (for example, W = F · s to find the work done by a constant force F in moving an object through a displacement s) have line integral versions that work for non-constant quantities (for example, W = ∫C F · ds to find the work done in moving an object along a curve C within a continuously varying gravitational or electric field F). A higher-dimensional analog of a line integral is a surface integral, where the (double) integral is taken over a two-dimensional surface instead of along a one-dimensional curve. Surface integrals can also be thought of as generalizations of multiple integrals. All of these can be seen as special cases of integrating a differential form, a viewpoint which allows multivariable calculus to be done independently of the choice of coordinate system. While the elementary notions upon which integration is based date back centuries before Newton and Leibniz independently invented calculus, line and surface integrals were formalized in the 18th and 19th centuries as the subject was placed on a rigorous mathematical foundation. The modern notion of differential forms, used extensively in differential geometry and quantum physics, was pioneered by Élie Cartan in the late 19th century.

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