Mathematical analysis

The Analysis [ analyzɪs ] (Greek ανάλυσις Analysis, Resolution ', ancient Greek ἀναλύειν analýein, resolve ') is a branch of mathematics, the foundations of Gottfried Wilhelm Leibniz and Isaac Newton were developed as infinitesimal calculus independently. The fundamental analysis is concerned with limits of sequences and series, as well as with functions of real numbers and their continuity, differentiability and integration. The methods of analysis are in all natural sciences and engineering is of great importance.

The generalization of the concept of function in calculus to functions of definition and target quantity in the complex numbers is part of the theory of functions.

Branches of Analysis

The Analysis has developed into a very general, not clearly definable generic term for a wide range of areas. In addition to the differential and integral calculus calculus includes other areas that build on it. These include the theory of ordinary and partial differential equations, the calculus of variations, vector analysis, the measure and integration theory and functional analysis.

One of its roots also has the function theory in analysis. Thus, the question of which functions satisfy the Cauchy -Riemann differential equations, be understood as a question of the theory of partial differential equations.

The opinion also the areas of harmonic analysis, differential geometry with subdivisions differential topology and global analysis, analytic number theory, the non-standard calculus, distribution theory and microlocal analysis can be counted in whole or in parts of it.

One-Dimensional Real Analysis

Differential calculus

For a linear function or a straight line

M is the slope and C is the y -axis or ordinate of the straight line portion. If you only have two points on a straight line, so the slope can be calculated by

For non-linear functions such as the gradient can thus no longer be calculated, since these curves described and thus are not straight lines. However, you can create a point a tangent again a straight line represents. The question now is how to calculate the slope of such a tangent at one point. If one chooses a point very close to and puts a line through the points and so the slope of this secant is almost the slope of the tangent. The slope of the secant (see above)

This quotient is called the difference quotient or average rate of change. If we ever closer to the point, we obtain by difference quotient, the slope of the tangent. We write

And call it the derivative or the derivative of f in. The term implies that x is always continued to align itself, or that the distance between x and is arbitrarily small. We also say: " x approaches ". The designation stands for Limes.

There are also cases in which this limit does not exist. That's why we introduced the concept differentiability. A function f is differentiable at the point when the limit exists.

Integral calculus

The calculus deals vividly with the calculation of areas under graphs of functions. This area may be approximated by a sum of partial surfaces and is in the limit into the integral.

The above sequence converges if f certain conditions (such as continuity ) is satisfied. This descriptive representation ( approximation by means of upper and lower sums ) corresponds to the so-called Riemann integral, which is taught in school.

In the so-called Higher Analysis moreover be more integral terms, such as viewing the Lebesgue integral.

Fundamental theorem of integral and differential calculus

Differential calculus and integral calculus behave according to the fundamental theorem of calculus in the following way "inverse" to each other.

If f is a continuous real function on a compact interval, then for:

And, if f is differentiable on uniformly continuous in addition,

Therefore, the set of all antiderivatives of a function is also called the indefinite integral and symbolized by.

Multidimensional Real Analysis

Many textbooks distinguish between analysis and in an analysis in several dimensions. This differentiation does not affect the basic concepts, but there are several dimensions in a larger mathematical diversity. The multidimensional analysis considered functions of several real variables, which are often represented as a vector or n - tuple.

The terms of the standard ( as a generalization of the amount ) of convergence, continuity and limits can be similarly generalized in several dimensions.

Important concepts from the multi-dimensional differential calculus are the direction and the partial derivative, which leads are in a variable or a direction. The set of black ensures when partial or directional derivatives of different directions may be reversed. In addition, the concept of total differentiation of meaning. This can be interpreted as a local adaptation of a linear mapping of the course of the multi-dimensional function and is multidimensional analogue of the ( one-dimensional ) derivative. The set of the implicit function of the local, unambiguous resolution of implicit equations is an important statement of the multidimensional analysis and can be used as a basis of differential geometry are understood.

In the multivariate analysis, there are different integral terms as the line integral, the surface integral and volume integral. However, from a more abstract point of view of vector analysis, these terms do not differ. To solve these integrals are the transform set as a generalization of the substitution rule and Fubini's theorem, which allows integrals over n-dimensional quantities converted into iterated integrals of particular importance. The integral theorems of vector analysis of Gauss, Green and Stokes are in the multidimensional analysis of meaning. They can be seen as a generalization of the fundamental theorem of integral and differential calculus.

Functional Analysis

The functional analysis is one of the most important areas of analysis. The key idea in the development of functional analysis was to develop a coordinate- and non-dimensional theory. This brought not only a formal income, but also enabled the study of functions on infinite-dimensional topological vector spaces. Here not only the real analysis and topology are linked together, but also methods of algebra play an important role. For important results of functional analysis as it is for example the set of Fréchet - Riesz, is central methods of the theory of partial differential equations can be derived. In addition, the functional analysis, in particular the spectral theory, the appropriate framework for the mathematical formulation of quantum mechanics and their uplifting theories.

Theory of differential equations

A differential equation is an equation containing an unknown function, and derivatives thereof. Stepping into the equation only ordinary derivatives, so called the differential equation usually. An example is the differential equation

Of the harmonic oscillator. From a partial differential equation is when partial derivatives appearing in the differential equation. An example of this class is the Laplace equation

The aim of the theory of differential equations is to find solutions, solution methods, and other properties of such equations. For ordinary differential equations, a comprehensive theory was developed, with which it is possible to specify to the given equations solutions exist inasmuch this. As partial differential equations are more complex in structure, there are a number of theories which can be applied to a large class of partial differential equations. Therefore, by examining the area of ​​partial differential equations generally only a single or small class of equations. Especially methods from functional analysis and also from the distribution theory and microlocal analysis can be used to find solutions and properties of such equations. However there are many partial differential equations, in which only limited information could be placed on the solution structure displayed by using these analytical methods. An important example in the physics of such a complex system, the partial differential equation is the Navier -Stokes equations. For this and for other partial differential equations, one tries to find approximate solutions in numerical mathematics.

Function theory

In contrast to real analysis, which deals only with functions of real variables (also called complex analysis ) are used in the theory of functions of complex variable functions were analyzed. The function theory has been withdrawn from the real analysis with independent methods and different issues. However, some phenomena of real analysis can be properly understood only with the help of function theory. Transferring issues of real analysis in the theory of functions can therefore lead to simplifications.