Differential

In functional analyzes and vectorial, one calls differential order 1 of a function in a point $a$ the linear part of the increase in this function between $a$ and $a\; +\; h$ when $h$ tends towards 0. It generalizes the calculation of Dérivée S and developments limited to the functions from several variables. This differential always does not exist, a function having a differential is called a differentiable function . One can then calculate differentials of a nature higher than 1.

One uses the differential notation with much effectiveness within the framework of the calculation of approximations and of the calculation of Dérivée S. It facilitates the formula of the derived from made up the. It appears very practical in the change of variable in Integral calculus.

In the approach of Leibniz, the differential of a function is its “infinitesimal increase”, which is written like a combination of the infinitesimal increases in the various variables, thus for a function $f$ of the variables $x$ and $y$

$\backslash \; mathrm\; df\; =\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}\; \backslash \; mathrm\; dx+\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; there\}\; \backslash \; mathrm\; dy=p\; \backslash ,\; \backslash \; mathrm\; dx+q\; \backslash ,\; \backslash \; mathrm\; dy$

Differential calculus thus conceived, if it were an effective computational tool, lacked a rigorous base. The differential fills this gap. Intuitively it is the algebraic tool which makes it possible to pass from the increases in variables $\backslash \; delta\; X,\; \backslash \; delta\; y$ with the increase in function $\backslash \; delta\; f$, at least with the first order of approximation. Mathematically it is not any more question of small variation but of calculation to the first order, whose definition is expressed in the form of a limit.

It is however advisable not to neglect the power of evocation and the effectiveness in calculations from the original point of view of Leibniz, it is what explains why there massively remains used, in particular by the physicists. By introducing the advanced concept of tensorial calculus on the varieties, the mathematicians could ensure a precise statute the differential notations of all kinds.

First approach

Function of only one variable

Differential calculus, for the functions of only one variable, merges with the Dérivation. That is to say $f$ a function of a real variable, with actual values; one will note $y=f\; (X)$ the result of the application of $f$. It is known as derivable when

$f\; (+\; H\; has)\; =\; F\; (a)\; +\; f\text{'}\; (A)\; \backslash ,\; H\; +h\; \backslash \; varepsilon\; (H)\; \backslash ,$

where $\backslash \; varepsilon$ is a function having a null limit into 0. One often summarizes that by the notation (often known as Notation of Pram)

$f\; (+\; H\; has)\; =\; F\; (a)\; +\; f\text{'}\; (A)\; \backslash ,\; H\; +o\; (H)\; \backslash ,$

Intuitively this calculation of limit, which bears the name of Développement limited to order 1 for the function $f$ in $a$, means that at first approximation, for $h$ near to 0, the expression $f\; (a+h)$ is not very different from the expression $f\; (a)\; +\; f\text{'}\; (a).h$. In particular among the expressions of the form $\backslash \; alpha+\; \backslash \; beta\; h$, it is this one which gives the best approximation of $f\; (a+h)$.

Intuitive introduction of the notations of the infinitesimal calculus

In many applications, speaking notations are employed to describe this situation. One agrees to note the $h$ number by $\backslash \; mathrm\; dx$ to indicate that it represents a very small variation of $x$ compared to the value of reference $a$. One notes $\backslash \; mathrm\; dy$ the variation of the image compared to the value of reference:

$\backslash \; mathrm\; dy=f\; (a+h)\; -\; F\; (a)\; \backslash \; simeq\; f\text{'}\; (A)\; \backslash \; mathrm\; dx\; \backslash ,$

The usually adopted point of view, abusive in any rigor, is that for sufficiently small variations, one can write $\backslash \; mathrm\; dy=f\text{'}\; (A)\; \backslash \; mathrm\; dx$. This presentation indeed retracts the need for using a calculation of limit, because even for very small variations, the term of error noted $o\; (H)$ above does not have a raison d'être null. Mathematically speaking it would be righter to note that

$\backslash \; delta\; there\; \backslash \; simeq\; f\text{'}\; (A)\; \backslash \; delta\; x$

Because the mathematicians prove the exact formula $\backslash \; mathrm\; dy=f\text{'}\; (A)\; \backslash \; mathrm\; dx$, while giving to the notations $\backslash \; mathrm\; dx$ and $\backslash \; mathrm\; dy$ a precise direction which is not that of small variations and which will be detailed lower.

Function of two variables

That is to say a function $f$ of the two variables $x$ and $y$; one will note $z=f\; (X,\; there)$ the result of the application of $f$.

Value awaited for the differential

Again the put question can be formulated as follows: when, compared to values of reference $a$ and $b$, one increases the variables $x$ and $y$ quantities $\backslash \; mathrm\; dx$ and $\backslash \; mathrm\; dy$, which is the effect (with the first order) on the variable $z$?

The partial derivative make it possible to answer the question when one of the two variations is null. Thus, because it is a simple calculation of derived from function of a variable, it is possible to write

$\backslash \; mathrm\; dz\; =\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}\; (has,\; b)\; \backslash \; mathrm\; dx\; \backslash \; text\; \{when\}\; \backslash \; mathrm\; dy=0$

and the same by reversing the roles: if $\backslash \; mathrm\; dx$ is null, $\backslash \; mathrm\; dz$ is calculated using the second partial derivative.

There would seem natural that when one increases $x$ and $y$ respectively quantities $\backslash \; mathrm\; dx$ and $\backslash \; mathrm\; dy$, the total increase is obtained by superimposing the two preceding cases

$\backslash \; mathrm\; dz\; =\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}\; (has,\; b)\; \backslash \; mathrm\; dx\; +\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\}\; (is,\; b)\; \backslash \; mathrm\; dy$

what in physique one in general states in the form: the total differential is the sum of the partial differential .

One will write for example: if $z\; =\; x^2\; +\; xy\; \backslash ,$ then $\backslash \; mathrm\; dz\; =\; (2x+y)\; \backslash ,\; \backslash \; mathrm\; dx\; +\; X\; \backslash ,\; \backslash \; mathrm\; dy$

In fact, this formula will be checked in very many explicit calculations; but it is not true in any general information.

The problem of the differentiability

It is necessary to detail the reasoning to see where it sins: one can make undergo initially an increase in $\backslash \; mathrm\; dx$ with the only variable $x$, which makes it pass from the value $a$ to $a+\; \backslash \; mathrm\; dx$, while $y$ remains equal to $b$. Then, by maintaining $x=a+\; \backslash \; mathrm\; dx$ constant, one makes pass $y$ of $b$ to $b+\; \backslash \; mathrm\; dy$. The resulting increases in $z$ are thus more precisely

$\backslash \; mathrm\; dz\_1\; =\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}\; (has,\; b)\; \backslash \; mathrm\; dx\; \backslash \; qquad\; \backslash \; hbox\; \{and\}\; \backslash \; qquad$

\ mathrm dz_2 = \ frac {\ partial F} {\ partial there} (a+ \ mathrm dx, b) \ mathrm Dy and still if this second quantity exists indeed.

The existence of derivative partial at the point $(has,\; b)$ is a priori insufficient to write a general formula of computation of $\backslash \; mathrm\; dz$. On the other hand, by adding conditions on the behavior of the derivative partial in the vicinity of $(has,\; b)$, one will be able to affirm indeed that $\backslash \; mathrm\; dz$ with the awaited value.

Definition of the differential

In general terms, the differentiability is the existence of a development limited to order 1 in a point, and the differential is the part of order 1 (thus linear) exactly.

For a real function with two variables

Initially let us study a function of two variables, with actual values: one will note $z=f\; (X,\; there)$. This function will be known as differentiable at the point $\backslash \; vec\; \{has\}$ coordinates $(X,\; there)$ if there exists a formula of limited development of order 1 for the function in this point, i.e.

$f\; (x+h,\; y+k)\; =\; F\; (X,\; there)\; +\; \backslash \; alpha\; h+\; \backslash \; beta\; K\; +\; O\; \backslash \; left\; (\backslash \; sqrt\; \{h^2+k^2\}\; \backslash \; right)$

with $\backslash \; alpha$ and $\backslash \; beta$ of the real coefficients.

If the function is differentiable, it is shown that the coefficients $\backslash \; alpha$ and $\backslash \; beta$ are well the derivative partial of $f$. One can then write

$f\; (x+h,\; y+k)\; =f\; (\backslash \; vec\; \{has\}\; +\; \backslash \; vec\; \{S\})\; =\; F\; (\backslash \; vec\; \{has\})\; +L\; (\backslash \; vec\; \{S\})\; +\; O\; (\backslash |\backslash \; vec\; \{S\}\; \backslash |)$

with the following expression which is linear in $\backslash \; vec\; \{S\}$

$L\; (\backslash \; vec\; \{S\})\; =\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}\; (X,\; there)\; \backslash \; cdot\; h+\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; there\}\; (X,\; there)\; \backslash \; cdot\; k$

The linear application $L$ is called differential $f$ to the point $\backslash \; vec\; \{has\}$ and noted $L=\; \backslash \; mathrm\; df\; (\backslash \; vec\; \{has\})$ .

One can take again the intuitive interpretation of $L$. If the variables undergo a small modification $\backslash \; vec\; \{H\}\; =\; \backslash \; vec\; \{\backslash \; delta\; \backslash \; ell\}$, the effect on the function is a modification $\backslash \; delta\; F\; =\; L\; (\backslash \; vec\; \{\backslash \; delta\; \backslash \; ell\})$, with the proviso of hastening to add: at least with the first order.

Note: the calculation of $L\; (\backslash \; vec\; \{S\})$ can also be presented like a scalar calculation of Produit with the vector gradient of $f$.

Generalizations in finished dimension

This first concept spreads with the functions of $\backslash \; R^n$ in $\backslash \; R$, by changing the number of variables simply, then with the functions of $\backslash \; R^n$ in $\backslash \; R^p$ by admitting vectorial coefficients for the limited development. A function $\backslash \; vec\; \{F\}$ of $\backslash \; R^n$ in $\backslash \; R^p$ will be known as differentiable in $\backslash \; vec\; \{has\}$ if there exists a development of the form

$\backslash \; vec\; \{F\}\; (a\_1+h\_1,\; \backslash \; dowries,\; a\_n+h\_n)\; =\; h\_1\; \backslash \; vec\; \{\backslash \; alpha\_1\}\; +\; \backslash \; dowries\; +\; h\_n\; \backslash \; vec\; \{\backslash \; alpha\_n\}\; +\; O\; \backslash \; left\; (\backslash |\backslash \; vec\; \{H\}\; \backslash |\backslash \; right)$

with $\backslash |\backslash \; vec\; H\; \backslash |$ which indicates the standard vector of components $(h\_1,\dots ,\; h\_n)$. Again, if the function is differentiable, it is shown that the coefficients $\backslash \; vec\; \{\backslash \; alpha\_1\}$ appearing in this development are the derivative partial of $\backslash \; vec\; \{F\}$. One will thus note

$\backslash \; mathrm\; D\; \backslash \; vec\; \{F\}\; (\backslash \; vec\; \{has\})\; (\backslash \; vec\; \{H\})\; =h\_1\; \backslash \; cdot\; \backslash \; frac\; \{\backslash \; partial\; \backslash \; vec\; \{F\}\}\; \{\backslash \; partial\; x\_1\}\; (\backslash \; vec\; \{has\})\; +\; \backslash \; dowries\; +h\_n\; \backslash \; cdot\; \backslash \; frac\; \{\backslash \; partial\; \backslash \; vec\; \{F\}\}\; \{\backslash \; partial\; x\_n\}\; (\backslash \; vec\; \{has\})$

To carry out this calculation it is judicious to introduce matric representations for the vector $\backslash \; vec\; \{H\}$ and the linear application $\backslash \; mathrm\; D\; \backslash \; vec\; \{F\}\; (\backslash \; vec\; \{has\})$: it is what one calls the Matrice jacobienne of the application, it is a matrix of dimension $(N,\; p)$.

It will be noted that, if the differentiability of the function ensures the existence of derivative partial, the reciprocal one is false: the existence of derivative partial always does not ensure the differentiability of the function.

There exists however a positive test: when the derivative partial of $f$ exist and are continuous, $f$ is differentiable.

General standard

More generally, it is possible to define the concept of differentiability and differential without having recourse to bases. Are $E$ and $F$ two normalized vector spaces, and $f$ an application of $E$ in $F$. That is to say $a$ a point of $E$. One gives up the notation of the vectors by arrows in this paragraph.

It is said that $f$ is differentiable in $a$ if and only if there exists a continuous linear application $L$ of $E$ in $F$ such as:

$\backslash \; forall\; H\; \backslash \; in\; E\; \backslash \; quad\; F\; (a+h)\; =f\; (a)+L\; (H)\; +o\; \backslash \; left\; (\backslash |H\; \backslash |\backslash \; right)$

In this case, $L$ is called differential $f$ in $a$ and notes $L=\; \backslash \; mathrm\; df\; (a)$.

As for the continuity of the linear applications, the diferentiability depends on the selected standard. One finds the usual definition in finished dimension since all the linear applications are continuous, all the equivalent standards.

Note: one can notice the semantic change between the first definition, that of Leibniz - a very small increase -, and that formalized nowadays - a linear application. This change is the result of an evolution of more than three centuries between an intuitive idea of the infinitesimal calculus and its formalization.

Differential of a higher nature

Case of the real function

If $y\; =\; F\; (X)$, if $f$ is Dérivable on $I$, then $\backslash \; mathrm\; df\; =\; f\text{'}\; (X)\; \backslash \; mathrm\; dx$. So moreover, $f\text{'}$ is derivable, $\backslash \; mathrm\; df$ is differentiable and

$\backslash \; mathrm\; d^2f\; =\; F$ (X) \; \mathrm dx\;\ mathrm dx. This quantity is called the differential of order 2 of $f$.

More generally, if $f$ is $n$ derivable time on $I$, one calls differential of order $n$ on $I$, the expression

$\backslash \; mathrm\; d^nf\; =\; f^\; \{(N)\}(X)\; (\backslash \; mathrm\; dx)\; ^n\; \backslash ,$

Case of the real function with two variables

If $f$ is a differentiable function on $I$ (open of $\backslash \; R^2$), then $\backslash \; mathrm\; df\; =\; \backslash \; tfrac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}\; \backslash \; mathrm\; dx\; +\; \backslash \; tfrac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; there\}\; \backslash \; mathrm\; dy$, each function $\backslash \; tfrac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}$ and $\backslash \; tfrac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\}$ is itself there a function of $\backslash \; R^2$ in $\backslash \; R$. If they are differentiable of continuous differential (i.e. $C^1$) then $\backslash \; mathrm\; df$ is also differentiable and

$\backslash \; mathrm\; d^2f\; =\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; ^2\; F\}\; \{\backslash \; partial\; X\; \backslash \; partial\; X\}\; \backslash \; mathrm\; dx+\; \backslash \; frac\; \{\backslash \; partial\; ^2\; F\}\; \{\backslash \; partial\; X\; \backslash \; partial\; there\}\; \backslash \; mathrm\; Dy\; \backslash \; right)\; \backslash \; mathrm\; dx+\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; ^2\; F\}\; \{\backslash \; partial\; there\; \backslash \; partial\; X\}\; \backslash \; mathrm\; dx+\; \backslash \; frac\; \{\backslash \; partial\; ^2\; F\}\; \{\backslash \; partial\; there\; \backslash \; partial\; there\}\; \backslash \; mathrm\; Dy\; \backslash \; right)\; \backslash \; mathrm\; dy$.

As the differentials are continuous, the Théorème of Schwarz makes it possible to say that:

$\backslash \; frac\; \{\backslash \; partial\; ^2\; F\}\; \{\backslash \; partial\; X\; \backslash \; partial\; there\}\; =\; \backslash \; frac\; \{\backslash \; partial\; ^2\; F\}\; \{\backslash \; partial\; there\; \backslash \; partial\; X\}$

what makes it possible to write the differential of order 2 of $f$ in the following form:

$\backslash \; mathrm\; d^2f\; =\; \backslash \; frac\; \{\backslash \; partial\; ^2\; F\}\; \{\backslash \; partial\; X\; \backslash \; partial\; X\}\; (\backslash \; mathrm\; dx)\; ^2\; +\; 2\; \backslash \; frac\; \{\backslash \; partial\; ^2\; F\}\; \{\backslash \; partial\; X\; \backslash \; partial\; there\}\; \backslash \; mathrm\; dx\; \backslash ;\; \backslash \; mathrm\; Dy\; +\; \backslash \; frac\; \{\backslash \; partial\; ^2\; F\}\; \{\backslash \; partial\; there\; \backslash \; partial\; there\}\; (\backslash \; mathrm\; Dy)\; ^2\; =\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; X\}\; \backslash \; mathrm\; dx\; +\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; there\}\; \backslash \; mathrm\; Dy\; \backslash \; right)\; ^2f$

where $\backslash \; bigl\; (\backslash \; tfrac\; \{\backslash \; partial\}\; \{\backslash \; partial\; X\}\; \backslash \; mathrm\; dx\; +\; \backslash \; tfrac\; \{\backslash \; partial\}\; \{\backslash \; partial\; there\}\; \backslash \; mathrm\; Dy\; \backslash \; bigr)$ becomes an operator acting on $f$.

More generally, if $f$ is of class $C^n$ then

$\backslash \; mathrm\; d^n\; F\; =\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; X\}\; \backslash \; mathrm\; dx\; +\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; there\}\; \backslash \; mathrm\; Dy\; \backslash \; right)\; ^n\; f$

General case

August 1st

See too

• differential Calculus

• differential Form
• Cohomologie

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