Mathematical analysis
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. The Greek mathematician Eudoxus also made explicit use of limits when he applied the same technique to calculate volumes. In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century CE to find the area of a circle. 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. Archimedes' work The Method of Mechanical Theorems rediscovered in the 20th century showed his explicit use of infinitesimals. These ancient methods laid the groundwork for later formal theories without yet having a rigorous definition of continuity.
Descartes's publication of La Géométrie in 1637 introduced the Cartesian coordinate system and is considered to be the establishment of mathematical analysis. Fermat developed analytic geometry which allowed him to determine the maxima and minima of functions and the tangents of curves. A few decades after Descartes, Newton and Leibniz independently developed infinitesimal calculus. This new field grew with the stimulus of applied work that continued through the 18th century into topics such as the calculus of variations. Mathematicians began applying calculus techniques to approximate discrete problems by continuous ones during this period. The independent development of these two giants created a powerful toolset that would eventually evolve into modern analysis.
Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816. Bolzano's work did not become widely known until the 1870s. 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. Cauchy formulated calculus in terms of geometric ideas and infinitesimals while introducing the concept of the Cauchy sequence. Around the same time, Weierstrass developed the epsilon-delta definition of limit approach thus founding the modern field of mathematical analysis. Dedekind constructed the real numbers by Dedekind cuts which formally defined irrational numbers to fill gaps between rational numbers. Lebesgue greatly improved measure theory and introduced his own theory of integration now known as Lebesgue integration around the early 20th century.
The vast majority of classical mechanics relativity and quantum mechanics is based on applied analysis and differential equations in particular. Newton's second law the Schrödinger equation and the Einstein field equations serve as important examples of differential equations used in physics. When processing signals such as audio radio waves light waves seismic waves and even images Fourier analysis can isolate individual components of a compound waveform. A large family of signal processing techniques consist of Fourier-transforming a signal manipulating the Fourier-transformed data in a simple way and reversing the transformation. Numerical linear algebra is important for data analysis while stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.
Modern numerical analysis
does not seek exact answers because exact answers are often impossible to obtain in practice. Much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors. Ordinary differential equations appear in celestial mechanics involving planets stars and galaxies. Stochastic differential equations help simulate living cells for medicine and biology applications. The study of algorithms that use numerical approximation distinguishes this field from discrete mathematics. In the 21st century life sciences and even the arts have adopted elements of scientific computations to solve complex problems that resist symbolic manipulation.
Common questions
Who used the method of exhaustion to compute the area inside a circle?
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. The Greek mathematician Eudoxus also made explicit use of limits when he applied the same technique to calculate volumes.
When was mathematical analysis established as an independent subject?
Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816. Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work in 1821.
What publication introduced the Cartesian coordinate system?
Descartes's publication of La Géométrie in 1637 introduced the Cartesian coordinate system and is considered to be the establishment of mathematical analysis. This event marked the beginning of analytic geometry which allowed mathematicians to determine the maxima and minima of functions.
How did Archimedes demonstrate his use of infinitesimals?
Archimedes' work The Method of Mechanical Theorems rediscovered in the 20th century showed his explicit use of infinitesimals. These ancient methods laid the groundwork for later formal theories without yet having a rigorous definition of continuity.
Why does modern numerical analysis seek approximate solutions instead of exact answers?
Modern numerical analysis does not seek exact answers because exact answers are often impossible to obtain in practice. Much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors.