Fundamental theorem of calculus

The Fundamental Theorem of Calculus, also known as the fundamental theorem of differential and integral calculus (HDI), brings the two basic concepts of Analysis with each other, namely, that of integration and differentiation. He indicates that deriving and integrating respectively the inversion of the other. The set consists of two parts, which are sometimes referred to as first and second fundamental theorem of calculus. The specific formulation of the theorem and its proof vary depending on the structure of the considered integration theory. Here, the Riemann integral is first considered.

• 6.1 The main theorem for Lebesgue integrals
• 6.2 The main theorem in the case of pointwise continuity
• 6.3 The main theorem in the complex
• 6.4 Multidimensional generalizations

History and reception

Already Isaac Barrow, the academic teacher Newton realized that surface charge ( integral calculus ) and tangent calculation ( calculus ) are in some sense inverses of each other, but the main sentence, he did not find. The first person who published this was 1667 James Gregory in Geometriae pars universalis. The first to recognize both the relationship as well as its fundamental importance, were independently Isaac Newton and Gottfried Wilhelm Leibniz and their calculus. In the first records for the Fundamental Theorem of 1666 Newton explains the theorem for arbitrary curves through the zero point, which is why he ignored the constant of integration. Newton published the Philosophiae Naturalis until 1686, Principia Mathematica. Leibniz took the phrase in 1677, he wrote it down substantially in today's notation.

Its modern form of the sentence received by Augustin Louis Cauchy, who gave a formal definition of first integral and a proof using the mean value theorem. Included in this is his continuation of Cours d'analyze of 1823. Cauchy also examined the situation in the complexes, thus demonstrating a number of key results of the theory of functions. During the 19th century it was the extensions to higher dimensions. Henri Léon Lebesgue advanced then in 1902 the fundamental theorem with the help of its Lebesgue integral on discontinuous functions.

The law was set to music in the 20th century in the main clause cantata.

The set

The first part of the theorem yields the existence of common features and the relationship between derivative and integral.

Be a real-valued continuous function on the closed interval, then for all the integral function

Differentiable and a primitive function to, that is, it applies to all. Note that the function due to the existence of the Riemann integral for continuous functions at all points in time is defined.

The second part of the sentence explains how integrals can be calculated.

Let be a continuous function with antiderivative, then applies the Newton - Leibniz formula:

The proof

The proof of the theorem is as soon as the terms derivative and integral are given, not difficult. The special achievement of Newton and Leibniz consists in the discovery of the statement and its relevance. For the first part only needs to be found that the discharge of, represented by, and there is the same.

In this regard it firmly and with. Then we have

By the mean value theorem of integral calculus, there exists a real number between and, so that

Applies. Because for and the continuity of it follows

I.e., the derivative of in and there is.

This part of the main theorem can be proved without the mean value theorem only by making use of the continuity.

The proof of the second part is done by inserting: If, for the given in the first part antiderivative, then and, and thus the theorem for this particular antiderivative. All other primitives differ from that but only by a constant, which disappears in the subtraction. Thus, the set of all common functions is proved.

Intuitive Explanation

To illustrate the explanation we consider a particle moving through space, described by the local function. The derivative of the position function with respect to time gives the velocity:

The spatial function is therefore an antiderivative of the velocity function. The main theorem now explains how the function itself can be recovered by integration of the derivative of a function. The above equation states that infinitesimal changes in time trigger an infinitesimal movement in the place:

A change in location is the sum of infinitesimal changes. However, these are given by the above equation, as the sum of the products of the derivative and infinitesimal changes in time. Exactly this process corresponds to the calculation of the integral of.

Applications

Calculation of integrals by primitives

The main significance of the fundamental theorem is that he attributes to the calculation of integrals to the determination of a primitive function, if such ever existed.

Examples

The defined on all function has a primitive function, and thus we obtain

The on -defined function whose graph describes the boundary of a unit semicircle, has the master function. For the area of the unit circle, one obtains the value

The last example shows how difficult it can be to root functions given functions easy to guess. Occasionally, this process extends the class of known functions. About the antiderivative of the function is not a rational function, but is related to the logarithm and is.

Derivation of integration rules

The relationship between integral and derivative allows derivation rules which can be easily proved from the definition of the derivative to transfer to integration rules on the law. For example, the power rule can be used to write down integrals of power functions directly. More interesting are statements that apply to more general classes of functions. It is then the transfer of the product rule, the partial integration, which is therefore also called product integration, and from the chain rule, the substitution rule. Only this provides a viable method for finding antiderivatives and thus the calculation of integrals.

Even with these possibilities and created in this way table works of master functions, however, there integrand, for non-root function can be specified, although the integral exists. The calculation shall then other tools of analysis carried out, for example, integration in the complex or numerically.

Generalizations of the main theorem

In its above form of the sentence only applies to continuous functions, which is a too strong restriction means. In fact, discontinuous functions such as the signum function can have a primitive function. For example, the rate also applies to the rule or Cauchyintegral be examined at the control functions. These have at any point in a left side and a right-side limit, which therefore can have very many points of discontinuity. This function class is still not sufficient, so here follows the main clause for the very general Lebesgue integral.

The main theorem for Lebesgue integrals

Is on Lebesgue integrable, then for any function

Absolutely continuous ( in particular, it is almost everywhere differentiable ), and it is -almost everywhere.

Conversely, suppose that the function is absolutely continuous. Then - almost everywhere differentiable. If we define as for all, where is differentiable, and identically zero for the other, it follows that the Lebesgue - integrable with

The main theorem in the case of pointwise continuity

Furthermore, the Fundamental Theorem of Calculus can also be formulated for functions that have only one point of continuity. For this purpose let Lebesgue integrable and continuous at the point. Then

Differentiable in, and it is. If or, the differentiability is one-sided to understand.

The main theorem in the complex

The main theorem can also be applied to contour integrals in the complex plane. Its importance lies in contrast to real analysis less in the statement itself and its importance for the practical calculation of integrals, but rather that follow from him three of the important sentences of the theory of functions, namely the Cauchy integral theorem and from then the Cauchy integral formula and the residue theorem. It is these records that are used for the calculation of complex integrals.

Let be a complex curve with parameter interval and a complex function on the open set which contains the conclusion of. Here, it is on and steady on the completion of complex differentiable. Then

In particular, this integral to zero, if a closed curve. The evidence leads to the integral simply real integrals of the real and imaginary parts back and uses the real fundamental theorem.

Multidimensional generalizations

Abstractly speaking, the value of an integral over an interval depends only on the values ​​of the strain at the edge function. This is generalized to higher dimensions by the Gaussian integral theorem, which brings the volume integral of the divergence of a vector field with an integral over the edge into connection.

It is compact with smooth boundary sections, the edge is oriented by an outward normal unit field, and also the vector field is continuous, and continuously differentiable in the interior of. Then we have

Even more broadly, the set of Stokes differential forms on manifolds. Be an oriented -dimensional differentiable manifold with smooth boundary sections with induced orientation. This is for the most vivid examples of how the full sphere with boundary ( sphere ), if. It should also be a continuously differentiable differential form of degree. Then we have

The Cartan derivative referred.