# Fundamental theorem of analysis

The fundamental theorem of analysis, also known as the main theorem of differential and integral calculus (HDI), is a mathematical theorem that connects the two basic concepts of analysis , namely that of integration and that of differentiation . It says that deriving or integrating is the opposite of the other. The main theorem of differential and integral calculus consists of two parts, sometimes referred to as the first and second law of differential and integral calculus . The concrete formulation of the theorem and its proof vary depending on the structure of the integration theory under consideration. Here the Riemann integral is considered first.

## History and reception

Already Isaac Barrow , the academic teacher Newton realized that area calculation (calculus) and tangent calculation (calculus) are inverse to each other in some ways, the law, however, he did not find. The first to publish this was James Gregory in Geometriae pars universalis in 1667 . The first to recognize both the connection and its fundamental importance were Isaac Newton and Gottfried Wilhelm Leibniz with their infinitesimal calculus . In the first notes on the fundamental theorem from 1666, Newton explained the theorem for any curve through the zero point , which is why he ignored the constant of integration. Newton only published this in 1686 in Philosophiae Naturalis Principia Mathematica . Leibniz found the sentence in 1677 and essentially wrote it down in today's notation.

The theorem received its modern form from Augustin Louis Cauchy , who was the first to develop a formal integral definition and a proof using the mean value theorem . This is contained in his continuation of the Cours d'Analyse from 1823. Cauchy also examined the situation in the complex and thus proved a number of central results of the theory of functions . In the course of the 19th century the extensions to higher dimensions were found. In 1902, Henri Léon Lebesgue extended the fundamental theorem to include discontinuous functions with the help of his Lebesgue integral.

In the 20th century, the mathematician Friedrich Wille set the main movement to music in the main movement cantata .

## The sentence

The first part of the theorem shows the existence of antiderivatives and the connection between derivative and integral:

If there is a real-valued continuous function on the closed interval , then the integral function is for all${\ displaystyle f \ colon [a, b] \ to \ mathbb {R}}$ ${\ displaystyle [a, b] \ subset \ mathbb {R}}$${\ displaystyle x_ {0} \ in [a, b]}$

${\ displaystyle F \ colon [a, b] \ to \ mathbb {R}}$ With ${\ displaystyle F (x) = \ int _ {x_ {0}} ^ {x} f (t) \, {\ rm {d}} t}$

differentiable and an antiderivative of , that is, holds for all . ${\ displaystyle f}$${\ displaystyle x \ in [a, b]}$${\ displaystyle F ^ {\ prime} (x) = f (x)}$

The fact that the integral function is defined over the whole interval follows from the fact that the Riemann integral exists for every continuous function. ${\ displaystyle F}$${\ displaystyle [a, b]}$

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

If there is a continuous function with an antiderivative , then the Newton-Leibniz formula applies: ${\ displaystyle f \ colon [a, b] \ to \ mathbb {R}}$${\ displaystyle F \ colon [a, b] \ to \ mathbb {R}}$

${\ displaystyle \ int _ {a} ^ {b} f (x) \, {\ rm {d}} x = F (b) -F (a)}$

## The proof

To explain the notation in the proof

The proof of the theorem is not difficult, once the terms derivative and integral are given. 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 shown that the derivative of , given by , exists and is equal . ${\ displaystyle F}$${\ displaystyle \ lim _ {h \ to 0} {\ frac {F (x + h) -F (x)} {h}}}$${\ displaystyle f (x)}$

Be firm and with . Then: ${\ displaystyle x \ in [a, b]}$${\ displaystyle h \ neq 0}$${\ displaystyle x + h \ in [a, b]}$

${\ displaystyle {\ frac {F (x + h) -F (x)} {h}} = {\ frac {1} {h}} \ left (\ int _ {x_ {0}} ^ {x + h} f (t) \, {\ rm {d}} t- \ int _ {x_ {0}} ^ {x} f (t) \, {\ rm {d}} t \ right) = {\ frac {1} {h}} \ int _ {x} ^ {x + h} f (t) \, {\ rm {d}} t}$

According to the mean value theorem of integral calculus, there is a real number between and , so that ${\ displaystyle \ xi _ {h}}$${\ displaystyle x}$${\ displaystyle x + h}$

${\ displaystyle \ int _ {x} ^ {x + h} f (t) \, {\ rm {d}} t = h \ cdot f (\ xi _ {h})}$

applies. Because of for and the continuity of it follows ${\ displaystyle \ xi _ {h} \ to x}$${\ displaystyle h \ to 0}$${\ displaystyle f}$

${\ displaystyle \ lim _ {h \ to 0} {\ frac {F (x + h) -F (x)} {h}} = \ lim _ {h \ to 0} f (\ xi _ {h} ) = f (x),}$

d. i.e., the derivative of in exists and is . This part of the main theorem can also be proved without the mean value theorem, only by using continuity. ${\ displaystyle F}$${\ displaystyle x}$${\ displaystyle f (x)}$

The proof of the second part is done by substitution: If one sets for the antiderivative given in the first part , then and and thus the theorem applies to this special antiderivative. All other antiderivatives differ from the one but only by a constant that disappears during subtraction. Thus the theorem is proven for all antiderivatives. ${\ displaystyle x_ {0} = a}$${\ displaystyle F (a) = 0}$${\ displaystyle F (b) = \ int _ {a} ^ {b} f (x) \, {\ rm {d}} x}$

## Clear explanation

For a clear explanation we consider a particle that moves through space, described by the position function . The derivative of the position function with respect to time gives the velocity: ${\ displaystyle x (t)}$

${\ displaystyle {\ frac {{\ rm {d}} x} {{\ rm {d}} t}} = v (t)}$

The position function is therefore an antiderivative of the velocity function . The main theorem now explains how the function itself can be recovered from the derivative of a function by integration. The above equation says that an infinitesimal change in time causes an infinitesimal change in location:

${\ displaystyle {\ rm {d}} x = v (t) {\ rm {d}} t}$

A change in location results from the sum of infinitesimal changes . According to the above equation, however, these are given as sums of the products of the derivative and infinitesimally small changes in time. The calculation of the integral of corresponds to exactly this process . ${\ displaystyle x}$${\ displaystyle {\ rm {d}} x}$${\ displaystyle v (t)}$${\ displaystyle v (t)}$

## Applications

### Calculation of integrals by antiderivatives

The main meaning of the fundamental theorem is that it reduces the computation of integrals to the determination of an antiderivative, if one exists at all.

#### Examples

• The completely defined function has the antiderivative . One thus obtains:${\ displaystyle \ mathbb {R}}$${\ displaystyle f (x) = x ^ {2}}$${\ displaystyle F (x) = {\ frac {x ^ {3}} {3}}}$
${\ displaystyle \ int _ {0} ^ {2} x ^ {2} \, {\ rm {d}} x = F (2) -F (0) = {\ frac {2 ^ {3}} { 3}} - {\ frac {0 ^ {3}} {3}} = {\ frac {8} {3}}}$
• The function defined on , the graph of which describes the edge of a unit semicircle, has the antiderivative${\ displaystyle I = [- 1,1]}$${\ displaystyle g (x) = {\ sqrt {1-x ^ {2}}}}$
${\ displaystyle G (x) = {\ frac {1} {2}} (\ arcsin x + x \ cdot {\ sqrt {1-x ^ {2}}})}$.
The value is thus obtained for the area of ​​half the unit circle
${\ displaystyle \ int _ {- 1} ^ {1} g (x) \, {\ rm {d}} x = G (1) -G (-1) = {\ frac {\ pi} {2} }}$,
thus the value for the area of ​​the whole unit circle .${\ displaystyle \ pi}$

The last example shows how difficult it can be to simply guess antiderivatives of given functions. Occasionally this process extends the class of known functions. For example, the antiderivative of the function is not a rational function , but is related to the logarithm and is . ${\ displaystyle {\ frac {1} {x}}}$${\ displaystyle \ ln \ left | x \ right |}$

### Derivation of integration rules

The connection between integral and derivative allows derivation rules , which can easily be inferred from the definition of the derivative, to be transferred to integration rules via the main clause. For example, the power rule can be used to write down integrals of power functions directly. Statements that apply to more general classes of functions are more interesting. Here, then results as 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 then does a practicable method for finding antiderivatives and thus for calculating integrals provide.

Even in tables of antiderivatives created with these possibilities and in this way, there are integrands for which no antiderivative can be specified, although the integral exists. The calculation must then be carried out using other analysis tools, for example integration in the complex or numerically .

## Generalizations of the main theorem

In its above form, the theorem is only valid for continuous functions, which means too strong a restriction. In fact, discontinuous functions can also have an antiderivative. For example, the theorem also applies to the rule or Cauchy integral, in which rule functions are examined. These have a left-hand and a right-hand limit value at each point, so they can have a large number of discontinuities. This function class is not yet sufficient either, so the main theorem for the very general Lebesgue integral follows here .

### The main theorem for Lebesgue integrals

Is to be integrated Lebesgue , then for all the function ${\ displaystyle f}$${\ displaystyle [a, b]}$ ${\ displaystyle x_ {0} \ in [a, b]}$

${\ displaystyle F (x) = \ int _ {x_ {0}} ^ {x} f \, {\ rm {d}} \ lambda}$ With ${\ displaystyle x \ in [a, b]}$

absolutely continuous (in particular, it can be differentiated almost everywhere ), and it is true almost everywhere. ${\ displaystyle F '(x) = f (x)}$ ${\ displaystyle \ lambda}$

Conversely, let the function on absolutely continuous. Then-is differentiable almost everywhere. If one defines as for all in which is differentiable and identical zero for the others , then it follows that Lebesgue is integrable with ${\ displaystyle F}$${\ displaystyle [a, b]}$${\ displaystyle F}$ ${\ displaystyle \ lambda}$${\ displaystyle f}$${\ displaystyle f (x) = F '(x)}$${\ displaystyle x \ in [a, b]}$${\ displaystyle F}$${\ displaystyle x \ in [a, b]}$${\ displaystyle f}$

${\ displaystyle F (b) -F (a) = \ int _ {a} ^ {b} f \, {\ rm {d}} \ lambda.}$

### The main theorem in the case of point-wise continuity

Furthermore, the fundamental theorem of analysis can also be formulated for functions that have only one place of continuity. To do this, be Lebesgue-integrable and continuous at the point . Then ${\ displaystyle f \ colon [a, b] \ to \ mathbb {R}}$${\ displaystyle y \ in [a, b]}$

${\ displaystyle F (x): = \ int _ {a} ^ {x} f (t) \, {\ rm {d}} t}$

in differentiable, and it holds . If or , the differentiability is to be understood one-sided. ${\ displaystyle y}$${\ displaystyle F ^ {\ prime} (y) = f (y)}$${\ displaystyle y = a}$${\ displaystyle y = b}$

### The main clause in the complex

The main clause can also be transferred to curve integrals in the complex number plane. In contrast to real analysis, its meaning lies less in the statement itself and its meaning for the practical calculation of integrals, but in the fact that it follows three of the important theorems of function theory, namely the Cauchy's integral theorem and from it then the Cauchy's integral formula and the Residual theorem . It is these theorems that are used to calculate complex integrals.

Let be a complex curve with a parameter interval and a complex function on the open set that contains the closure of . Here is complex differentiable on and continuously on the conclusion of . Then ${\ displaystyle \ gamma}$${\ displaystyle [a, b]}$${\ displaystyle F}$${\ displaystyle U}$${\ displaystyle \ gamma}$${\ displaystyle F}$${\ displaystyle U}$${\ displaystyle \ gamma}$

${\ displaystyle \ int _ {\ gamma} F '(z) \, {\ rm {d}} z = F (\ gamma (b)) - F (\ gamma (a)).}$

In particular, this integral is zero when is a closed curve. The proof simply reduces the integral to real integrals of the real part and the imaginary part and uses the real main theorem. ${\ displaystyle \ gamma}$

### Multi-dimensional generalizations

In abstract terms, the value of an integral on an interval depends only on the values ​​of the antiderivative at the edge. This is generalized to higher dimensions by the Gaussian integral theorem, which connects the volume integral of the divergence of a vector field with an integral over the boundary. ${\ displaystyle \ mathrm {F}}$

It should be compact with sections smooth edge , the edge is oriented by an external field normal unit , further, the vector field is continuous on and continuously differentiable in the interior of . Then: ${\ displaystyle V \ subset \ mathbb {R} ^ {n}}$ ${\ displaystyle S}$${\ displaystyle {\ rm {n}}}$${\ displaystyle {\ rm {F}}}$${\ displaystyle V}$${\ displaystyle V}$

${\ displaystyle \ int _ {V} \ operatorname {div} {\ rm {F}} \; {\ rm {d}} V = \ oint _ {\ partial V} {\ rm {F}} \ cdot { \ rm {d}} {\ rm {S}}}$

More generally, Stokes's theorem considers differential forms on manifolds. Let be an oriented -dimensional differentiable manifold with a smooth edge in sections with induced orientation . This is the case for most illustrative examples, such as the full sphere with a rim (sphere). Furthermore, let it be a continuously differentiable differential form of degree . Then applies ${\ displaystyle M}$${\ displaystyle n}$ ${\ displaystyle \ partial M}$${\ displaystyle \ omega}$${\ displaystyle n-1}$

${\ displaystyle \ int _ {M} {\ rm {d}} \ omega = \ int _ {\ partial M} \ omega,}$

where denotes the Cartan derivative . ${\ displaystyle {\ rm {d}}}$