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## 3 Benefits of Learning the History of Mathematics

Mathematical analysis, a branch of mathematics that deals with the establishment and techniques of differential and integral calculus, and other examples of the use of the limit (or threshold crossings) as a theory of (infinite) lines, products infinities, the development of analytic functions in order, analytic extension, variation account and others.

Historical development

The differential and integral calculus founded by Newton and Leibniz was driven by geometry (tangent to the curve, area under the curve) and mechanics (mainly the velocity-particle relationship). Their calculation, however, is not rigorous, and only comes close to the intuitive notion of limit (limit) compared to the unsubstantiated mathematical introduction of infinitesimal size.

Development of the concept of limit values

In clumsy hands, maneuvering with infinitely small sizes easily leads to a contradiction. Only Cauchy in the 19th century arrives at a rigorous notion of limit (definition “epsilon-delta”), which ultimately founds logic and calculation and thus provides the foundation for modern (mathematical) analysis. This avoids having to account for infinitely small and manipulated quantities with the limits calculated from the final small sizes. The advantage of working with finite sizes is also a disadvantage, because the idea of ​​infinitely small size and very intuitively easily transferable to engineers, but accessible via the epsilon-delta technique. To this end, in the middle of the 20th century, the logician Abraham Robinson founded a new approach, called. non-standard analysis in which the infinitely small and infinitely large size rigorously imposed, and the rules of these accounts, which leads to a contradiction. In the non-standard analysis with single elements, “standards” of the set of real numbers are introduced and the so-called. non-standard items. In addition to statements that only talk about the standard elements of the non-standard and introduced testimonials in which they occur and the non-standard elements. Robinson introduced the principle of transfer or forwarding which allows standard expressions derived from non-standard statements to be extended.

Development of concepts and measures of integrals

Integral calculus of functions of real variables in the axis of the interval, based on the Riemann of the 19th century, based on the limits of Cauchy terms applied to the so-called. partial sums. Files are taken by interval divisions (subdivision) of the argument function over sub-intervals whose maximum length tends to zero. Such an integral is applied to a class of functions for which this limit exists, for example: integrable functions on Riemann or Riemann-integrable functions. At the beginning of the 20th century, Lebesgue observed another type of limit, where the subdivision instead of the argument, looking at the value of the subdivision, resulted in a successive approximation of the so-called integral sum. steep functions. This process leads to a more flexible notion of integrable which leads to the so-called. Lebesgue-integrable functions, and the development of the modern theory of abstract generalizations of the length and area of ​​geometric sets, now called action theory.

Functional analysis

Functional analysis is an examination of infinite dimensional spaces, a typical functional space. Functional is a function defined over (typically) infinite dimensional spaces of mappings (these areas alone are often not linear spaces). The most frequently used locally convex topological vector spaces, and among them were historically important Hilbert, Banach, and Frechet spaces.

The calculus of variations consists in finding the local extreme functions (minimums and maximums).

Function theory

Function theory deals with the properties and relationships of various classes of functions of one or more real or complex variables. These are the typical properties of mathematical analysis: measurability, il extremum, different types of integrable, analytic, finite variation, continuity, Lipschitz continuity, the existence of statements (derivation), the maximum principle, the existence and properties of limits on the boundary of area definitions, properties with respect to various integral transformations (Mellina, Fourier, Hilbert, Laplace…), asymptotic properties, analytic extension, zeros, etc. How often is the class of linear space functions interesting from the point of view of functional analysis, this field is intimately linked to functional analysis.

Harmonic analysis

Harmonic analysis comes from studies of normal modes and the oscillating wave system. It is very important in physics and engineering applications is important when the linear regime, for example in the theory of linear systems and the linear response of nonlinear systems. Mathematically, harmonic analysis consists in finding a certain group of dual symmetry, which allows, with certain structures of the theory of action, the development of a linear function of the corresponding bases in the functional spaces. This includes, for example, Fourier series and Fourier integrals. Noncommutative harmonic analysis deals with the decomposition of the functional space which is a key representation of continuous groups of symmetries, when these symmetries are sufficiently numerous (Frechet, when the so-called valid. Plancherel’s theorem). The representations are analogues of the one-dimensional functional subspace of the commutative case.

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