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## 47Analysis

The term analysis has had a shifting meaning during the development of mathematics. Around 1600, coming from ancient origins, the term was used to refer to calculations in which one worked with an unknown variable (e.g. “$x$”). The purpose of the calculations was connected to a concrete problem, e.g. finding a length. In other words, the term analysis was closely related to the term algebra and its meaning were to import algebraic notation to geometry. This was done by Descartes and others. However, over the course of the 18th century, the word became an own meaning and came to be associated with the calculus, i.e. a new branch of mathematics using special analytic techniques. These techniques included differential and integral calculus and rival methods for these techniques were devised by Newton and Leibniz the third quarter of the 17th century. Both synthesized and extended a considerable amount of earlier work dealing with normals and tangents to curves. These techniques proved to be notably successful and were extended in a variety of directions, in particular in mechanics and differential equations.

The key common feature of calculus is to combine infinitely many infinitely small (i.e. infinitesimal) quantities to get a finite answer. As an example, suppose we want to calculate the area of a circle. Doing it using analytical techniques, we might divide the circle into segments and approximate the area of each segment with the area of a triangle. As the number of segments gets higher and higher, the sum of the areas of all triangles will approximate the area of the circle. The following figures demonstrate the idea of creating infinitely many vanishingly small triangles to get the (finite) total area of a circle.

The introduction of techniques involving reasoning with infinitely small objects, limiting processes, infinite sums, fractions approaching $\frac 00,$ and so forth meant that the founders of calculus were exploring new ground. At the beginnings of calculus, their arguments were not very consistent and it took another decades and centuries to get rid of vague terminology and found the techniques on more solid and rigorous axiomatic basis.

In BookOfProofs, the branch of analysis is divided into different thematic parts, e.g.:

• Number systems, providing a foundation of numbers and arithmetics.
• Real Analysis of One Variable, addressing basic analytical concepts for the calculus of one real variable.
• Real Analysis of Multiple Variables, addressing basic analytical concepts for the calculus of more than one real variable.
• Complex Analysis, dealing with complex numbers and also known as the theory of functions::
• Differential Equations, dealing with equations containing an unknown function as a derivative and providing a theory capable to describe many physical phenomena,
• Linear Integral Equations, dealing with equations containing the unknown function under the integral
• Functional Analysis, discussing modern theories of differentiation and integration and the principal problems and methods of handling integral and linear functionals
• Vector Analysis, concerned with differentiation and integration of vector fields
• Calculus of Variations, which is concerned with the problem of “extremising” functionals.

Some of these parts are still under development and need to be extended. If you are a specialist in one of these areas, we are happy if you join our project to provide content to these parts.

| | | | Contributors: bookofproofs | References: [641]

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