Mathematical analysis
Mathematical analysis formally developed in the 17th century during the Scientific Revolution,[3] but many of its ideas can be traced back to earlier mathematicians.
Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids.
[5] The explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems, a work rediscovered in the 20th century.
[6] In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century CE to find the area of a circle.
[7] From Jain literature, it appears that Hindus were in possession of the formulae for the sum of the arithmetic and geometric series as early as the 4th century BCE.
[9] Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere in the 5th century.
Fermat's method of adequality allowed him to determine the maxima and minima of functions and the tangents of curves.
[13] Descartes's publication of La Géométrie in 1637, which introduced the Cartesian coordinate system, is considered to be the establishment of mathematical analysis.
It would be a few decades later that Newton and Leibniz independently developed infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations, ordinary and partial differential equations, Fourier analysis, and generating functions.
In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, particularly by Euler.
He also introduced the concept of the Cauchy sequence, and started the formal theory of complex analysis.
Poisson, Liouville, Fourier and others studied partial differential equations and harmonic analysis.
The contributions of these mathematicians and others, such as Weierstrass, developed the (ε, δ)-definition of limit approach, thus founding the modern field of mathematical analysis.
Around the same time, Riemann introduced his theory of integration, and made significant advances in complex analysis.
Towards the end of the 19th century, mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof.
Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions.
The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis.
Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence.
Most precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers.
Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics.
[21][22][23] Differential equations play a prominent role in engineering, physics, economics, biology, and other disciplines.
Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space or time (expressed as derivatives) is known or postulated.
This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time value varies.
Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement.
Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors.
Ordinary differential equations appear in celestial mechanics (planets, stars and galaxies); numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.
