Differential of a function
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A differential (or differential ) referred to in the analysis of the linear portion of the growth of a variable or a function. Historically, the term in the 17th and 18th centuries, the core of the development of calculus. From the 19th century, the analysis was rebuilt by Augustin Louis Cauchy mathematically correct and Karl Weierstrass on the basis of the limit concept, and the concept of differential lost for the elementary differential and integral calculus in importance.
If there is a functional dependency with a differentiable function, then is the fundamental relationship between the derivative of the dependent variable and the differential of the independent variable
Where the derivative of at the point referred to, instead of to write in this situation or not. This relationship can be generalized to functions of several variables using partial derivatives and then leads to the concept of total differential.
Differentials are today used in various applications in different meanings and also with different mathematical rigor. The differentials occurring in standard notation as for integrals or derivatives are now usually viewed as a mere notation constituent without independent significance.
A rigorous definition supplies the differential geometry, where differentials are interpreted as exact 1-forms. A different type of access provides the non-standard analysis, which takes up the historical concept of the infinitesimal again and clarified in the sense of modern mathematics.
- Blaise Pascal 5.1 Reflections on the quarter circle: Quarts de Cercle 5.1.1 similarity
- 6.1 constant and constant factor
- 6.2 Addition and Subtraction
- 6.3 multiplication
- 6.4 Division
- 6.5 Chain Rule
- 6.6 Examples
- 7.1 Notations of the derivative
- 7.2 Extension and variants 7.2.1 Totales differential
- 7.2.2 Virtual displacement variation derivation
- 7.2.3 vectorial nabla differential operator
- 7.2.4 Stochastic Analysis
The differential growth as a linearized
Is a real function of a real variable, thus causing a change of the argument to a change of the function value of at; for the growth of the function value therefore applies
For example, an ( affine ) a linear function, that is about as follows. That is, the growth of the function value is directly proportional to the increase of the argument in this simple case and the ratio corresponds exactly to the constant slope of.
For functions whose slope is not constant, the situation is more complicated. Is differentiable at the point, the gradient is given by where the derivative, which is defined as the limiting value of the differential quotients:
Referring now to the difference between the difference quotient of the derivative and
As follows for the representation
In this view, the growth of the function value is decomposed into a component, which depends linearly on, and a radical which disappears from higher than linear order for, that is true in the sense. The linear portion of the increase, which for small values of generally represents therefore a good approximation is called the differential of and is denoted by.
Definition
It is a function with domain of definition. Is differentiable at the point and then is
The differential at the point of the argument growth. Applies so we also write instead of.
Especially for the identical function, ie valid for the equation and thus
Analog can be defined higher order differentials: Is in the place - times differentiable () and is called so
The differential -th order at the point of the argument growth. In this product is referred to the -th derivative of the position and the -th power of the number.
Differentials in the integral calculus
To calculate the area of a region which is, and enclosed by the graph of a function of the axis and of two perpendicular straight lines, divided to the area into rectangles of the width, which are made "infinitely narrow", and height. Their respective area is the "product"
The entire surface area is, the sum
Here again with a finite size, which corresponds to a subdivision of the interval. See more precisely, mean value theorem of integral calculus. There is a fixed value in the interval whose function value is multiplied by the sum of the interval of finite value of the integral of this represents a continuous function:
The total interval of the integral need not be divided evenly. The differentials at different subdivision sites may be selected to be different in size, the choice of dividing the integration interval will often depend on the nature of the integration problem. Together with the function value within the " differential " interval (or the maximum and minimum value in it according to upper and lower sum ) to form a surface area; to make the limit transition in the sense that one chooses the subdivision of finer and finer. The integral is a definition for a surface with boundary by a curve segment.
As an example, the surface can serve under the parabola x ² in the interval 0 to 1, in which one elementary solves the Gaussian sum rate.
Order of the differentials
The differentials can also be easily and clearly represent in their order, but differently according to their dependence. So stands for the second order differential (corresponding second derivative ) of the independent variable and means ( in the definition which is really the square of differences), and ( as and written ) for the second order differential of the dependent variable, which is not the square of the differential, but rather the differential again formed from a differential (as defined by the difference of the difference). For this differential the calculation rules below apply.
Declaration of the second order differential
If we now chosen somehow, and that the same value for different, so held, so is a function of and can be a differential form ( see figure ) from her again.
For this differential can formally write. If and the same value for different is, this size is the linear portion of the growth, that is, it is. This applies accordingly for higher derivatives, eg.
Historical
Gottfried Wilhelm Leibniz first used in a manuscript in 1675 in the essay Analysis tetragonistica the integral sign, but he does not write. On November 11, 1675 Leibniz wrote an essay entitled " Examples of the inverse tangent method " and here comes before addition for the first time, instead of just the spelling.
In the modern version of this approach to integral calculus after Bernhard Riemann " integral " is a limit of the areas of finitely many rectangles of finite width for finer and finer subdivisions of the "region ".
Therefore, the first symbol is a stylized S in the integral for " sum ". Leibniz writes in 1675: " Utile erit scribi per omnia " ( It will be useful instead of writing omnia ). Omnia stands for omnia l and is used in the geometrically oriented surface calculation method of Bonaventura Cavalieri. The corresponding print publication Leibniz 's De geometria recondita of 1686th Leibniz gave himself with the notation trouble, " calculus moderately easy to make and inevitably to the bill. "
Blaise Pascal's reflections on the quarter circle: Quarts de Cercle
As Leibniz as a young man in 1673 in Paris was, he received an important stimulus from a consideration of Pascal in his Traité in 1659 published writing the sine of quarts de cercle (Treatise on the sine of the quarter circle ). He says he has seen in a light that the author did not notice. It is (written in modern terminology, see figure) to the following:
To the static moment
To determine the quarter circular arc with respect to the x-axis, Pascal excludes the similarity of the triangles with vertices
And
That behave like
And thus
So that
Applies. Leibniz noticed now - and this was the " light" which he saw - that this procedure is not limited to the circle, but generally applicable to all (smooth ) curve, where the circle of radius a by the length of the curve normals ( the reciprocal of the curvature, the radius of curvature of the circle ) is replaced. The infinitesimal triangle
Is the characteristic triangle ( It is also found in Isaac Barrow to the tangent determination. ) It is noteworthy that the later Leibnizian symbolism of the differential calculus (dx, dy, ds) just corresponds to the position of these " improved Indivisibilienvorstellung ".
Similarity
All the triangles of a portion of the tangent along with the respective x -and y- axis parallel lengths and form with the triangle of curvature radius a, subnormal and ordinate y similar triangles and keep the conditions in accordance with the slope of the tangent to the circle of curvature at this point, even when when the threshold crossing is taken. The ratio of is exactly the slope of. Therefore, it can transmit its ( characteristic ) proportions in the coordinate system on the differentials where each circle of curvature at a point of the curve, in particular if they are regarded as an infinitesimal size.
Nova methodus 1684
New method of maxima, minima and tangents, which pushes himself neither broken nor to irrational quantities, and a peculiar fact relevant invoice type. ( Leibniz ( G. G. L.), Acta eruditorum 1684)
Leibniz described here very briefly on four sides his method. He can select an independent firm differential ( here dx, see Fig ro) and returns the arithmetic rules as below, for the differentials to, describes how to make them.
Then he gives the chain rule:
This is unusual from today's perspective, because it requires independent and dependent differentials equal and individually, and not as final, considered the differential quotient of dependent and independent variable. The other way round, when he gives a solution, the formation of the derivative is possible. It covers the entire range of rational functions. This is followed by a formal complicated example, a dioptric refraction of light (minimum) [ anm. 1], and an easily solvable geometric, with intricate spacing ratios of [ anm. 2], and one that handles the logarithm.
Other relationships are scientifically, historically at him out of the context of earlier and later work on the subject, some of which exist only in manuscript or in letters and not made public. In Nova methodus 1684 is not, for example, that for the independent dx dx = const applies. and ddx = 0 In other contributions, he treats the topic to " roots " and quadratures of infinite series.
The ratio of Unendlichklein and known differential ( = size) describes Leibniz:
For the transcendent line the cycloid is used as proof.
As an appendix he declared in 1684, the solution of a problem that Florimond de Beaune Descartes asked, and he did not solve. The problem requires that a function (f, the line WW in Table XII) is found whose tangent (WC), the X- axis always intersects the portion between the intersection of the tangent with the x axis and the distance of which to the corresponding abscissa x, there he chooses dx always equal to b, constant, he calls it here a, is. This proportionality he compares to the arithmetic and geometric series, and as an abscissa and as ordinate the logarithms of the Numbers. "So it will be the ordinates w" ( increase in value ) " dw " ( pitch increase ), " their increments or differences, proportional, ..." He gives the logarithm function as a solution: " ... if the w the Numbers are, so are the x logarithms ". w = a / b dw or w dx = a dw. This satisfies
Or
Cauchy differential term
In the 1980s, a debate was held in Germany, the extent to which the foundation of the analysis in Cauchy is logically impeccable. Detlef Laugwitz tried using a historical reading of the Cauchy to make the concept of infinitely small quantities for his numbers fertile, but will, as a result in Cauchy discrepancies. Detlef corrects the gap ( ersten! ) historical Lesansatz Cauchy's work and calls for the use of terms from Cauchy's time and not today's terms as proof of their sentences and comes to the conclusion that the foundation of the Cauchy Analysis is logically perfect, but still the questions remain after treatment of infinitely small sizes open.
The differentials in Cauchy are finite and constant ( finally). The value of the constant is not defined.
Is infinitely small and variable in Cauchy.
The relationship is, with finite and infinitesimal ( infinitely small).
It's geometrical ratio is defined as
Determined. This ratio of infinitely small quantities, or more precisely the limit of geometrical differences ratios dependent numerical quantities, a quotient Cauchy can be transferred to finite sizes.
Differentials are finite number of variables whose geometric relationships are strictly equal to the limits of the geometric relationships that are formed from the infinitely small increments of the independent variable or variables of the functions presented. Cauchy considers it important to look at differentials as finite number of sizes.
The computer uses the infinity small as referring agents, which must lead him to the knowledge of the relationship that exist between the finite number of variables; and by Cauchy's opinion may the small infinity in the final equations, where their presence would remain meaningless, pointless and useless, never be allowed. Besides, if you looked at the differentials to be stable very small number of variables, then one would thus have the advantage of is that you can take as a unit under the differentials of several variables. Because in order to form a clear idea of any number size, it is important to refer to the unity of its genus. It is therefore important to select a unit under the differentials.
In particular, falls for the Cauchy problem away, to define higher differentials. For Cauchy sets after receiving the calculation rules of the differentials by the transition to the limits. And since the differential of a function of the variables is another function of these variables, it can differentiate several times and receives in this way the differentials of different orders.
Calculation rules
Regardless of the definition used for differentials apply the following calculation rules. Below we denote the independent variable, dependent variable and functions, and an arbitrary real constant. The derivation of after being written. Then the following calculation rules arising from the relationship
And the derivation rules.
Constant and constant factor
- And
- In particular, for the Independent Leibniz chooses the unit by setting:
Addition and subtraction
- And
Multiplication
Division
Chain rule
- Is dependent on and, therefore, and then applies
Examples
- Applies or for and. It follows
- For and and, so
Notations
Notations of the derivation
Or
(read: stick by)
(read: Two stick by)
Or
Or
Or
Extension and variants
Rather than see the following symbols denote the differentials:
- With ( derived from the Greek Delta, speaking, del ) is called a partial differential. The sign is a "round" fracture -d similar.
- With ( the Greek small delta ) is a virtual displacement, the variation of the position vector is called. So is related to the partial differential of the individual spatial dimensions of the position vector.
Total differential
A mixed form of various differentials and a sum instead of an integral character has the total differential or total differential:
The partial differentials are obtained by setting to be viewed equal to one, all the others equal to zero. If, for example, n = 2, then, after the partial differential by setting and. Whether can amalgamate the partial differentials of a total, always requires a separate clarification.
Virtual displacement variation derivation
The spatial coordinates of the physical system are Mathematically speaking at a virtual displacement at a fixed time t varies. The variation of the position vector, ie, the virtual displacement of the relevant system point
The symbol is therefore used for the variation of dissipation.
, The vector is regarded as a function. Likewise, the individual derivatives are determined according to the directions by above sets a equal to one and all others equal to zero.
Vectorial nabla differential operator
The nabla operator is a symbol of operation, which is used in the vector analysis to denote the three differential operators gradient, divergence and rotation. He is denoted by the nabla symbol ( or even to emphasize the formal similarity to conventional vector magnitudes ).
Stochastic Analysis
In the stochastic analysis the differential notation is frequently used as notation for stochastic differential equations; it is then always be understood as shorthand for a corresponding equation of Ito integrals. For example, a stochastic process with respect to a Wiener process Ito - integrable, then by
Given equation for a process in differential form as listed. The above calculation rules for differentials are, however, to modify, in the case of stochastic processes with non-vanishing quadratic variation according to the lemma of Itō.
Example: Mathematical Physics
As an example, serve an important first step of mathematical (or theoretical ) physics. This works as an integration theory with differentials.
Suppose that the second Newtonian axiom.
The composition does not depend on T, the movement takes place only on the x- axis.
The force is given as a linear force field and is considered as a function of location.
Is extended with dx / dt ( = v), a differential quotient:
We use the product rule on what is multiplied top left with m.
On the left is after this transformation a complete differential!
Then integrating both sides over t. Right is transferred to x ( the distance), as it is dx = (dx / dt) dt:
Links emphasizes the integral of the differential speed square on.
The kinetic energy is left only with different speeds equal to zero, what is the right definition of work:
The negative work A is the potential energy V, the lifting work. The potential occurs in different functional dependence, depending on how the force law reads. In this definition, an additive constant occurs, which shall be added and the total energy. (Eg: lifting at perigee with mgh to 0 meters or 100 meters)
Integrating with respect to time, when F is given as a function of t gives a definition of the pulse and its modification, as well as the Weggleichung kinematically because cancels out the mass.