Derivative and differential of a complex function of several variables. Derivatives of complex functions of several variables Derivative of a complex function of several derivatives
Let the function z - f(x, y) be defined in some domain D on the xOy plane. Let's take an interior point (x, y) from the region D and give x an increment Ax such that the point (x + Ax, y) 6 D (Fig. 9). Let's call the value a partial increment of the function z with respect to x. Compose the ratio For a given point (x, y), this ratio is a function of Definition. If for Ax -* 0 the relation ^ has a finite limit, then this limit is called the partial derivative of the function z = /(x, y) with respect to the independent variable x at the point (x, y) and is denoted by the symbol jfc (or /i(x, jj ), or z "x (x, In the same way, by definition, or, which is the same, Analogously If and is a function of n independent variables, then Noting that Arz is calculated with the value of the variable y unchanged, and Atz with the value of the variable x unchanged, definitions of partial derivatives can be formulated as follows: Partial derivatives The geometric meaning of partial derivatives of a function of two variables Differentiability of a function of several variables The necessary conditions function differentiability Sufficient conditions differentiability of functions of several variables Complete differential. Partial Differentials Derivatives of the compound function of the partial derivative with respect to x of the function z = /(x, y) is called the ordinary derivative of this function with respect to x, calculated under the assumption that y is a constant; the partial derivative with respect to y of a function z - /(x, y) is its derivative with respect to y, calculated under the assumption that x is a constant. This implies that the rules for calculating partial derivatives coincide with the rules proved for a function of one variable. Example. Find partial derivatives of a function 4 We have Substitutions*. The existence of a function y = /(x, y) at a given point of partial derivatives with respect to all arguments does not imply the continuity of the function at this point. So, the function is not continuous at the point 0(0,0). However, at this point specified function has partial derivatives with respect to x and y. This follows from the fact that /(x, 0) = 0 and /(0, y) = 0, and therefore the geometric meaning of the partial derivatives of a function of two variables. Let the surface S in three-dimensional space be given by the equation where f(x, y) is a function, continuous in some domain D and having partial derivatives with respect to x and y there. Let us find out the geometric meaning of these derivatives at the point Mo(x0, y0) 6 D, to which the point f(x0)yo) corresponds on the surface z = f(x)y). When finding the partial derivative at the point M0, we assume that z is only a function of the argument x, while the argument y retains a constant value y \u003d yo, i.e. The function fi (x) is geometrically represented by the curve L, along which the surface S is intersected by the plane y \u003d at about. Due to the geometric meaning of the derivative of a function of one variable, f \ (xo) = tg a, where a is the angle formed by the tangent to the line L at the point JV0 with the Ox axis (Fig. 10). But so Thus, the partial derivative ($|) is equal to the tangent of the angle a between the Ox axis and the tangent at the point N0 to the curve obtained in the section of the surface z \u003d / (x, y) by the y plane. Similarly, we obtain that §6. Differentiability of a Function of Several Variables Let the function z = /(x, y) be defined in some domain D on the xOy plane. Let us take a point (x, y) € D and give the chosen values x and y any increments Ax and Dy, but such that the point. Definition. A function r = /(x, y) is called a differentiable * point (x, y) € 2E if the total increment of this function, corresponding to the increments Dx, Dy of the arguments, can be represented as where A and B do not depend on Dx and D y ( but in general they depend on x and y), while a(Ax, Dy) and f(Ax, Dy) tend to zero as Ax and Dy tend to zero. . If the function z = /(x, y) is differentiable at the point (x, y), then the part A Dx 4 - VDy of the increment of the function, linear with respect to Dx and Dy, is called the total differential of this function at the point (x, y) and is denoted by the symbol dz: Tanim way, example. Let r = x2 + y2. At any point (r, y) and for any Dx and Dy we have Here. it follows that a and /3 tend to zero as Ax and Dy tend to zero. By definition, given function is differentiable at any point in the xOy plane. Here, we note that in our reasoning we did not formally exclude the case when the increments Dx, Dy separately, or even both at once equal to zero. Formula (1) can be written more compactly if we introduce the expression (the distance between the points (Using it, we can write Denoting the expression in brackets by e, we will have where c depends on J, Du and tends to zero if J 0 and Dy 0, or, in short, if p 0. Formula (1), which expresses the condition for the function z = f(xt y) to be differentiable at the point (x, y), can now be written as So, in Example 6.1 above. Theorem 4. If the function r = f(x, y) is differentiable at some point, then it is continuous at that point.4 If the function r = f(x, y) is differentiable at the point (x, y), then the total the increment of the function i at this point""e, corresponding to the increments j and dy of the arguments, can be represented as Theorem b. If the function r = f(x, y) is differentiable at a given point, then mo o u has partial derivatives at this point. Let the function z = /(x, y) be differentiable at a point (x, y). Then the increment Dx of this function, which corresponds to the increments Dx, Ay of the arguments, can be represented in the form (1). Taking in equality (1) Dx Ф 0, Dn \u003d 0, we get from where Since on the right side of the last equality the value A does not depend on, This means that at the point (x, y) there is a partial derivative of the function r \u003d / (x, y) with respect to x, and by similar reasoning we see that (x, there is a partial derivative of the function zу, and it follows from the theorem that We emphasize that Theorem 5 asserts the existence of partial derivatives only at the point (x, y), but does not say anything about their continuity 6.2 Sufficient Conditions for the Differentiability of Functions of Several Variables As is well known, a necessary and sufficient condition for the differentiability of a function y = f(x) of one variable at the point xo is the existence of a finite the derivative /"(x) at the point x0. In the case when the function depends on several variables, the situation is much more complicated: there are no necessary and sufficient conditions for differentiability for the function z = /(x, y) of two independent variables x, y; there is l look for the necessary conditions (cf. above) and separately - sufficient. These sufficient conditions for the differentiability of functions of several variables are expressed by the following theorem. Theorem c. If a function has partial derivatives /£ and f"v in some neighborhood of the thin line (xo, y0) and if these derivatives are continuous at the point (xo, y0) itself, then the function z = f(x, y) is differentiable at the point (x- Example Consider a function Partial derivatives Geometric meaning of partial derivatives of a function of two variables Differentiability of a function of several variables Necessary conditions for differentiability of a function Sufficient conditions for differentiability of functions of several variables Total differential Partial differentials Derivatives of a complex function It is defined everywhere Based on the definition of partial derivatives, we have ™ of this function at the point 0(0, 0) we find and the increment of this sharpens. 0 and Du 0. We put D0. Then from formula (1) we will have Therefore, the functions / (x, y) \u003d is not differentiable at the point 0 (0, 0), although it has at this point we produce fa and f "r Obtained the result is explained by the fact that the derivatives f"z and f"t are discontinuous at the point of §7. full differential. Partial differentials If the function r - f(z> y) is differentiable, then its last differential dz is their increments: After that, the formula for the total differential of the function takes the example. Let i - 1l(x + y2). Then Similarly, if u =) is a differentiable function of n independent variables, then Expression is called the lean differential of the function z = f(x, y) with respect to the variable x; the expression is called the partial differential of the function z = /(x, y) of the variable y. It follows from formulas (3), (4) and (5) that the total differential of a function is the sum of its partial differentials: Note that the total increment Az of the function z = /(x, y), generally speaking, is not equal to the sum of partial increments. If at a point (x, y) the function z = /(x, y) is differentiable and the differential dz Φ 0 at this point, then its total increment differs from its linear part only by the sum of the last terms aAx 4 - /? 0 and Ay --> O are infinitesimals of a higher order than the terms of the linear part. Therefore, when dz Ф 0, the linear part of the increment of a differentiable function is called the main part of the increment of the function and an approximate formula is used that will be the more accurate, the smaller the absolute value of the increments of the arguments. §eight. Derivatives of a complex function 1. Let the function be defined in some domain D on the xOy plane, and each of the variables x, y, in turn, is a function of the argument t: We will assume that when t changes in the interval (the corresponding points (x, y) do not go out outside the domain D. If we substitute the values into the function z = / (x, y), then we obtain a complex function of one variable t. and for the corresponding values the function / (x, y) is differentiable, then the complex function at the point t has a derivative and M Let us give t an increment Dt. Then x and y will receive some increments Ax and Dy. As a result, when (J)2 + (Dy)2 ∆ 0, the function z will also receive some increment Dt, which, due to the differentiability of the function z = /(x , y) at the point (x, y) can be represented as where a) tend to zero as Ax and Du tend to zero. We extend the definition of a and /3 for Ax = Ay = 0 by setting a Then a( will be continuous for J = Dy = 0. Consider the relation for the given one are constant, by condition there are limits from the existence of derivatives ^ and at the point £ it follows that the functions x = y(t) and y = are continuous at this point; therefore, at At 0 both J and Dy tend to zero, which in turn entails a(Ax, Dy) and P(Ax, Ay) tend to zero. Thus, the right-hand side of equality (2) at 0 has a limit equal to Hence, the limit of the left-hand side of (2) exists at At 0, i.e., exists equal Passing in equality (2) to the limit as At -» 0, we obtain the required formula In the particular case when, consequently, z is a complex function of x, we obtain y) over x, in the calculation of which in the expression /(x, y) the argument y is taken as a constant. independent variable x, in the calculation of which y in the expression /(x, y) is no longer taken as a constant, but is in turn considered a function of x: y = tp(x)t and therefore the dependence of z on x is taken into account completely. Example. Find and jg if 2. Consider now the differentiation of a complex function of several variables. Let where in turn so that Suppose that at the point (() there are continuous partial derivatives u, 3? and at the corresponding point (x, y), where The function f(x, y) is differentiable. Let us show that under these conditions the complex funchion z = z(() y) at the point t7) has derivatives and u, and we find expressions for these derivatives. Note that this case does not differ significantly from the one already studied. Indeed, when z is differentiated with respect to £, the second independent variable rj is taken as a constant, as a result of which x and y become functions of the same variable x" = c), y = c) in this operation, and the question of the derivative Φ is solved in exactly the same way as the question of derivative in the derivation of formula (3) Using formula (3) and formally replacing the derivatives g and ^ in it by the derivatives u and respectively, we obtain If a complex function is “Specified by formulas so that, if the appropriate conditions are met, we have In the particular case when And = where Partial derivatives Geometric meaning of partial derivatives of a function of two variables Differentiability of a function of several variables Necessary conditions for the differentiability of a function Sufficient conditions for the differentiability of functions of several variables Complete differential. Partial differentials We have derivatives of a complex function Here m is the total partial derivative of the function and with respect to the independent variable x, taking into account the complete dependence of and on x, including and through z = z(x, y), a
Differentiation complex functions
Let for the function n- variable arguments are also functions of variables:
The following theorem on differentiation of a compound function is valid.
Theorem 8. If the functions are differentiable at the point , and the function is differentiable at the corresponding point , where , . Then the complex function is differentiable at the point , and the partial derivatives are determined by the formulas
where the partial derivatives are calculated at the point and are calculated at the point .
ƒ Let us prove this theorem for a function of two variables. Let , a .
Let and be arbitrary increments of the arguments and at the point . They correspond to increments of functions and at the point . The increments and corresponds to the increment of the function at the point . Since it is differentiable at the point , its increment can be written as
where and are calculated at the point , at and . Due to the differentiability of the functions and at the point , we obtain
where is calculated at the point ; .
We substitute (14) into (13) and rearrange the terms
Note that as , since and tend to zero as . This follows from the fact that infinitesimal at and . But the functions and are differentiable and, therefore, continuous at the point . Therefore, if and , then . Then and at .
Since the partial derivatives are calculated at the point , we get
Denote
and this means that it is differentiable with respect to the variables and , and
Consequence. If , and , , i.e. , then the derivative with respect to the variable t calculated by the formula
If , then
The last expression is called total derivative formula for a function of many variables.
Examples. 1) Find the total derivative of the function , where , .
Decision.
2) Find the total derivative of the function if , .
Decision.
Using the rules of differentiation of a complex function, we obtain one important property of the differential of a function of many variables.
If the independent variables are functions, then the differential is by definition equal to:
Now let the arguments be differentiable functions at some point of the function with respect to the variables , and let the function be differentiable with respect to the variables , . Then it can be considered as a complex function of variables , . By the previous theorem, it is differentiable and the relation holds
where is determined by formulas (12). We substitute (12) into (17) and, collecting the coefficients at , we obtain
Since the coefficient of the derivative is equal to the differential of the function , then formula (16) was again obtained for the differential of the complex function.
Thus, the first differential formula does not depend on whether its arguments are functions or whether they are independent. This property is called invariance of the form of the first differential.
Taylor formula (29) can also be written as
ƒ The proof will be carried out for a function of two variables or .
Let's first consider a function of one variable. Let times be differentiable in a neighborhood of the point . The Taylor formula for a function of one variable with a remainder term in the Lagrange formula has
Since is an independent variable, then . By definition of the differential of a function of one variable
If we denote , then (31) can be written as
Consider some neighborhood of a point and an arbitrary point in it and connect the points and a straight line segment. It is clear that the coordinates and points of this line are linear functions of the parameter .
On the straight line segment, the function is a complex function of the parameter , since . Moreover, it is times differentiable with respect to and Taylor formula (32) is valid for, where , i.e.,
The differentials in formula (32) are the differentials of the complex function , where , , , i.e.
Substituting (33) into (32) and taking into account that , we obtain
The last term in (34) is called the remainder of the Taylor formula in Lagrange form
We note without proof that if, under the assumptions of the theorem, the function is differentiable at a point m times, then the remainder term can be written as Peano form:
Chapter 7
7.1. Space R n . Sets in linear space.
A set whose elements are all ordered sets from n real numbers, denoted and called n-dimensional arithmetic space, and the number n called dimension of space. The element of the set is called a point in space, or a vector, and the numbers coordinates this point. The point =(0, 0, …0) is called zero or origin.
Space is the set of real numbers, i.e. - number line; and are the two-dimensional coordinate geometric plane and the three-dimensional coordinate geometric space, respectively. The vectors , , …, are called single basis.
For two elements of a set, the concepts of the sum of elements and the product of an element by a real number are defined:
Obviously, by virtue of this definition and the properties of real numbers, the equalities are true:
According to these properties, the space is also called linear (vector) space.
In linear space is defined scalar product elements and as a real number calculated according to the following rule:
The number is called vector length or the norm. Vectors and are called orthogonal, if . Value
, )= │ - │ =
called spacing between elements and .
If and are nonzero vectors, then angle between them is called an angle such that
It is easy to verify that for any elements and real number, the scalar product is performed:
A linear space with a scalar product defined in it by formula (1) is called euclidean space.
Let point and . The set of all points for which the inequalities hold
called n -measuring cube with an edge and centered at the point . For example, a two-dimensional cube is a square with a side centered at .
The set of points satisfying the inequality are called n-ball radius centered at , which is also called
- the neighborhood of the point in and denote ,
Thus, a one-dimensional ball is an interval of length . 2D ball
there is a circle for which the inequality
Definition 1. The set is called limited, if exists
n is a ball containing this set.
Definition 2. A function defined on the set of natural numbers and taking values belonging to is called sequence in space and is denoted by , where .
Definition 3. The point is called sequence limit, if for an arbitrary positive number there exists a natural number such that the inequality holds for any number.
Symbolically, this definition is written as follows:
Designation:
It follows from Definition 3 that , for . Such a sequence is called converging to .
If the sequence does not converge to any point, then it is called divergent.
Theorem 1. For the sequence to converge to a point it is necessary and sufficient that for any number , i.e. to sequence i- x coordinates of the points converged to i-th coordinate of the point .
□ The proof follows from the inequalities
The sequence is called limited, if the set of its values is limited, i.e.
Like a number sequence, a convergent sequence of points is bounded and has a single limit.
Definition 4. The sequence is called fundamental(Cauchy sequence), if for any positive number one can specify a natural number such that for arbitrary natural numbers and greater than , , i.e.
Theorem 2(Cauchy criterion). For a sequence to converge, it is necessary and sufficient that it be fundamental.
□ Necessity. Let converge to a point . Then we get a sequence converging to . . . , …, X is called region in . If a X - area, then its closure is called closed area.
Sets X and Y called separable, if none of them contains touchpoints of the other.
A bunch of X called related if it cannot be represented as a union of two separable sets.
A bunch of X called convex , if any two of its points can be connected by a segment that entirely belongs to this set.
Example. Based on the definitions above, it can be argued that
– connected, linearly connected, open, non-convex set, is a region.
– connected, linearly connected, non-open, non-convex set, is not a domain.
– unconnected, not linearly connected, open, non-convex set, is not a region.
– unconnected, not linearly connected, open set, not a domain.
– connected, linearly connected, open set, is a domain.
Theorem.Let be u = f(x, y) is given in the domain D and let x = x(t) and y = y(t) defined in the area , and when , then x and y belong to the area D. Let a function u be differentiable at a point M 0 (x 0 ,y 0 ,z 0), and functions x(t) and at(t) are differentiable at the corresponding point t 0 , then the complex function u = f[x(t),y(t)]=F (t)differentiable at t 0 and the following equality holds:
.
Proof. Since u is conditionally differentiable at the point ( x 0 , y 0), then its total increment is represented as
Dividing this ratio by , we get:
Let us pass to the limit at and obtain the formula
.
Remark 1. If a u= u(x, y) and x= x, y= y(x), then the total derivative of the function u by variable X
or 
.
The last equality can be used to prove the rule for differentiating a function of one variable given implicitly in the form F(x, y) = 0, where y= y(x) (see topic number 3 and example 14).
We have: 
. From here 
. (6.1)
Let's go back to example 14 of topic number 3:
;
.
As you can see, the answers are the same.
Remark 2. Let be u = f (x, y), where X= X(t , v), at= at(t , v). Then u is ultimately a complex function of two variables t and v. If now the function u is differentiable at a point M 0 (x 0 , y 0), and the functions X and at are differentiable at the corresponding point ( t 0 , v 0), then we can talk about partial derivatives with respect to t and v from a complex function at a point ( t 0 , v 0). But if we are talking about the partial derivative with respect to t at the specified point, then the second variable v is considered constant and equal to v 0 . Therefore, we are talking about the derivative only of a complex function with respect to t and, therefore, we can use the derived formula. Thus, we get.
The proof of the formula for the derivative of a complex function is given. Cases where a complex function depends on one or two variables are considered in detail. A generalization is made to the case of an arbitrary number of variables.
ContentSee also: Examples of applying the formula for the derivative of a complex function
Basic formulas
Here we present the derivation of the following formulas for the derivative of a complex function.
If , then
.
If , then
.
If , then
.
Derivative of a complex function of one variable
Let a function of a variable x be represented as a complex function in the following form:
,
where and there are some functions. The function is differentiable for some value of the variable x . The function is differentiable for the value of the variable .
Then the complex (composite) function is differentiable at the point x and its derivative is determined by the formula:
(1)
.
Formula (1) can also be written as follows:
;
.
Proof
Let us introduce the following notation.
;
.
Here there is a function of variables and , there is a function of variables and . But we will omit the arguments of these functions so as not to clutter up the calculations.
Since the functions and are differentiable at the points x and , respectively, then at these points there are derivatives of these functions, which are the following limits:
;
.
Consider the following function:
.
For a fixed value of the variable u , is a function of . It's obvious that
.
Then
.
Since the function is a differentiable function at the point , then it is continuous at that point. So
.
Then
.
Now we find the derivative.
.
The formula has been proven.
Consequence
If a function of variable x can be represented as a complex function of a complex function
,
then its derivative is determined by the formula
.
Here , and there are some differentiable functions.
To prove this formula, we sequentially calculate the derivative according to the rule of differentiation of a complex function.
Consider a complex function
.
Its derivative
.
Consider the original function
.
Its derivative
.
Derivative of a complex function in two variables
Now let a complex function depend on several variables. First consider case of a complex function of two variables.
Let the function depending on the variable x be represented as a complex function of two variables in the following form:
,
where
and there are differentiable functions for some value of the variable x ;
is a function of two variables, differentiable at the point , . Then the complex function is defined in some neighborhood of the point and has a derivative, which is determined by the formula:
(2)
.
Proof
Since the functions and are differentiable at the point , they are defined in some neighborhood of this point, are continuous at the point, and their derivatives at the point exist, which are the following limits:
;
.
Here
;
.
Due to the continuity of these functions at a point, we have:
;
.
Since the function is differentiable at the point , it is defined in some neighborhood of this point, is continuous at this point, and its increment can be written as follows:
(3)
.
Here
- function increment when its arguments are incremented by the values and ;
;
- partial derivatives of the function with respect to the variables and .
For fixed values of and , and there are functions of the variables and . They tend to zero as and :
;
.
Since and , then
;
.
Function increment :
.
:
.
Substitute (3):
.
The formula has been proven.
Derivative of a complex function of several variables
The above derivation is easily generalized to the case when the number of variables of a complex function is greater than two.
For example, if f is function of three variables, then
,
where
, and there are differentiable functions for some value of the variable x ;
is a differentiable function, in three variables, at the point , , .
Then, from the definition of differentiability of the function , we have:
(4)
.
Since, due to continuity,
;
;
,
then
;
;
.
Dividing (4) by and passing to the limit , we obtain:
.
And finally, consider the most general case.
Let a function of a variable x be represented as a complex function of n variables in the following form:
,
where
there are differentiable functions for some value of the variable x ;
- differentiable function of n variables at a point
,
,
... , .
Then
.
