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1 LECTURE N Total differential, partial derivatives and differentials of higher orders Total differential Partial differentials Partial derivatives of higher orders Higher order differentials 4 Derivatives of complex functions 4 Total differential Partial differentials If a function z=f(,) is differentiable, then its total differential dz is equal to dz= a +B () z z Noting that A=, B =, we write the formula () in the following form z z dz= + () We extend the concept of a function differential to independent variables, setting the differentials of independent variables equal to their increments: d= ; d= After that, the formula for the total differential of the function will take the form z z dz= d + d () d + d n variables, then du= d (d =) = The expression d z=f (,)d (4) is called the partial differential of the function z=f(,) with respect to the variable; the expression d z=f (,)d (5) is called the partial differential of the function z=f(,) with respect to the variable It follows from formulas (), (4) and (5) that the total differential of a function is the sum of its partial differentials: dz=d z+d z the increment z= z z + + α (,) + β (,) differs from its linear part dz= z z + only by the sum of the last terms α + β, which at 0 and 0 are infinitesimal higher order than the terms of the linear part Therefore for dz 0, the linear part of the increment of the differentiable function is called the main part of the increment of the function and the approximate formula z dz is used, which will be the more accurate, the smaller the absolute value of the increments of the arguments,97 Example Calculate approximately arctg(),0

2 Solution Consider the function f(,)=arctg() Using the formula f(x 0 + x, y 0 + y) f(x 0, y 0) + dz, we get arctg(+) arctg() + [ arctg() ] + [ arctg()] or + + arctg() arctg() () + () Let =, =, then =-0.0, =0.0 Therefore, (0.0 0.0 arctg) arctg( ) + (0.0) 0.0 = arctan 0.0 = + 0.0 + () + () π = 0.05 0.0 0.75 4 It can be shown that the error resulting from the application of the approximate formula z dz does not exceed the number = M (+), where M is the largest value of the absolute values ​​of the second partial derivatives f (,), f (,), f (,) when the arguments change from to + and from to + Partial derivatives of higher orders If the function u =f(, z) has a partial derivative with respect to one of the variables in some (open) domain D, then the found derivative, being itself a function of, z, can, in turn, have partial derivatives at some point (0, 0, z 0) with respect to the same or any other variable For the original function u=f(, z), these derivatives will be partial derivatives of the second order If the first derivative was taken, for example ep, in, then its derivative with respect to, z is denoted as follows: f (0, 0, z0) f (0, 0, z0) f (0, 0, z0) = ; = ; = or u, u, u z z z Derivatives of the third, fourth, and so on orders are determined similarly. Note that the higher-order partial derivative taken with respect to various variables, for example, ; called mixed partial derivative Example u= 4 z, then, u =4 z ; u = 4z; u z = 4 z; u = z u=64z; uzz = 4; u = z u = z u z = 4 z; u z =8 z; u z =6 4 z; u z =6 4 z the function f(,) is defined in an (open) domain D,) in this domain there are first derivatives f and f, as well as second mixed derivatives f and f, and finally,) these last derivatives f and f, as functions of u, are continuous in some point (0, 0) of the region D Then at this point f (0, 0)=f (0, 0) Proof Consider the expression

3 f (0 +, 0 f (0 +, 0) f (0, 0 + f (0, 0) W=, where, are non-zero, for example, are positive, and, moreover, are so small that D contains the entire rectangle [ 0, 0 +; 0, 0 +] 0 +) (, 0) ()= and therefore continuous With this function f (0 +, 0 f (0 +, 0) f (0, 0 f (0, 0) expression W, which is equal to W= can be rewritten in the form: ϕ (0 +) ϕ (0) W= so: W=ϕ (0 + θ, 0 f (0 + θ, 0) (0 + θ)= (0<θ<) Пользуясь существованием второй производной f (,), снова применим формулу конечных приращений, на этот раз к функции от: f (0 +θ,) в промежутке [ 0, 0 +] Получим W=f (0 +θ, 0 +θ), (0<θ <) Но выражение W содержит и, с одной стороны, и и, с другой, одинаковым образом Поэтому, можно поменять их роли и, введя вспомогательную функцию: Ψ()= f (0 +,) f (0,), путем аналогичных рассуждений получить результат: W=f (0 +θ, 0 +θ) (0<θ, θ <) Из сопоставления () и (), находим f (0 +θ, 0 +θ)=f (0 +θ, 0 +θ) Устремив теперь и к нулю, перейдем в этом равенстве к пределу В силу ограниченности множителей θ, θ, θ, θ, аргументы и справа, и слева стремятся к 0, 0 А тогда, в силу (), получим: f (0, 0)=f (0, 0), что и требовалось доказать Таким образом, непрерывные смешанные производные f и f всегда равны Общая теорема о смешанных производных Пусть функция u=f(, n) от переменных определена в открытой n-мерной области D и имеет в этой области всевозможные частные производные до (n-)-го порядка включительно и смешанные производные n-го порядка, причем все эти производные непрерывны в D При этих условиях значение любой n-ой смешанной производной не зависит от того порядка, в котором производятся последовательные дифференцирования Дифференциалы высших порядков Пусть в области D задана непрерывная функция u=f(, х), имеющая непрерывные частные производные первого порядка Тогда, du= d + d + + d

4 We see that du is also a function of, If we assume the existence of continuous partial derivatives of the second order for u, then du will have continuous partial derivatives of the first order and we can talk about the total differential of this differential du, d(du), which is called second-order differential (or second differential) of u; it is denoted by d u We emphasize that the increments d, d, d are considered constant and remain the same when passing from one differential to the next (moreover, d, d will be zero) So, d u=d(du)=d(d + d + + d) = d() d + d() d + + d() d or d u = (d + d + d + + d) d + + (d + d + = d + d + + d + dd + dd + + dd + + Similarly, the third-order differential d u is defined, and so on. If the function u has continuous partial derivatives of all orders up to and including the nth one, then the existence of the nth differential is guaranteed. But the expressions for them become more and more complex We can simplify the notation Let's take out the "letter u" in the expression of the first differential Then, the notation will be symbolic: du=(d + d + + d) u ; d u=(d + d + + d) u ; d n n u=(d + d + + d) u, which should be understood as follows: first, the “polynomial” in brackets is formally raised to a power according to the rules of algebra, then all the resulting terms are “multiplied” by u (which is added to n in the numerators at) , and only after that all symbols return their value as derivatives and differentials u d) d u on the variable t in some interval: =ϕ(t), =ψ(t), z=λ(t) Let, in addition, as t changes, the points (, z) do not go beyond the region D Substituting the values, and z into function u, we get a complex function: u=f(ϕ(t), ψ(t), λ(t)) Suppose that u has continuous partial derivatives u, u and u z in and z and that t, t and z t exist Then it is possible to prove the existence of a derivative of a complex function and calculate it. We give the variable t some increment t, then, and z will receive increments, respectively, and z, the function u will receive an increment u Let us represent the increment of the function u in the form: (this can be done, since we assumed the existence of continuous partial derivatives u, u and u z) u=u +u +u z z+α +β +χ z, where α, β, χ 0 at, z 0 We divide both part of the equality on t, we obtain u z z = u + u + uz + α + β + χ t t t t t t t 4

5 Let us now let the increment t approach zero: then, z will tend to zero, since the functions, z of t are continuous (we assumed the existence of derivatives t, t, z t), and therefore, α, β, χ also tend to zero In the limit we obtain u t =u t +u t +u z z t () We see that under the assumptions made, the derivative of the complex function does exist. If we use the differential notation, then du d d dz () will look like , z in several variables t: =ϕ(t, v), =ψ(t, v), z=χ(t, v) Besides the existence and continuity of partial derivatives of the function f(, z), we assume here the existence of derivatives of functions, z with respect to t and v This case does not differ significantly from the one already considered, since when calculating the partial derivative of a function of two variables, we fix one of the variables, and we are left with a function of only one variable, the formula () will be the same z, and () must be rewritten as: = + + (a) t t t z t z = + + (b) v v v z v Example u= ; =ϕ(t)=t ; =ψ(t)=cos t u t = - t + ln t = - t- ln sint 5

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Consider changing a function when incrementing only one of its arguments − x i, and let's call it .

Definition 1.7.private derivative functions by argument x i called .

Designations: .

Thus, the partial derivative of a function of several variables is actually defined as the derivative of the function one variable - x i. Therefore, all the properties of derivatives proved for a function of one variable hold true for it.

Comment. In the practical calculation of partial derivatives, we use the usual rules for differentiating a function of one variable, assuming that the argument with respect to which differentiation is carried out is variable, and the remaining arguments are constant.

1. z= 2x² + 3 xy –12y² + 5 x – 4y +2,

2. z = x y ,

Geometric interpretation of partial derivatives of a function of two variables.

Consider the surface equation z = f(x,y) and draw a plane x = const. Let us choose a point on the line of intersection of the plane with the surface M (x, y). If you set the argument at increment Δ at and consider the point T on the curve with coordinates ( x, y+Δ y, z+Δy z), then the tangent of the angle formed by the secant MT with the positive direction of the O axis at, will be equal to . Passing to the limit at , we obtain that the partial derivative is equal to the tangent of the angle formed by the tangent to the resulting curve at the point M with the positive direction of the O axis y. Accordingly, the partial derivative is equal to the tangent of the angle with the O axis X tangent to the curve resulting from the section of the surface z = f(x,y) plane y= const.

Definition 2.1. The full increment of the function u = f(x, y, z) is called

Definition 2.2. If the increment of the function u \u003d f (x, y, z) at the point (x 0, y 0, z 0) can be represented in the form (2.3), (2.4), then the function is called differentiable at this point, and the expression is called the main linear part of the increment or the total differential of the function under consideration.

Notation: du, df (x 0 , y 0 , z 0).

Just as in the case of a function of one variable, the differentials of independent variables are their arbitrary increments, therefore

Remark 1. Thus, the statement "the function is differentiable" is not equivalent to the statement "the function has partial derivatives" - differentiability also requires the continuity of these derivatives at the point under consideration.

4. Tangent plane and normal to the surface. The geometric meaning of the differential.

Let the function z = f(x, y) is differentiable in a neighborhood of the point M (x 0, y 0). Then its partial derivatives are the slopes of the tangents to the lines of intersection of the surface z = f(x, y) with planes y = y 0 and x = x 0, which will be tangent to the surface itself z = f(x, y). Let's write an equation for the plane passing through these lines. The direction vectors of the tangents have the form (1; 0; ) and (0; 1; ), so the normal to the plane can be represented as their vector product: n = (- ,- , 1). Therefore, the equation of the plane can be written as:

where z0 = .

Definition 4.1. The plane defined by equation (4.1) is called tangent plane to the graph of the function z = f(x, y) at the point with coordinates (x 0, y 0, z 0).

From formula (2.3) for the case of two variables it follows that the increment of the function f in the vicinity of the point M can be represented as:

Therefore, the difference between the applicates of the function graph and the tangent plane is an infinitesimal higher order than ρ, at ρ→ 0.

In this case, the differential of the function f looks like:

which corresponds increment of the applicate of the tangent plane to the graph of the function. This is the geometric meaning of the differential.

Definition 4.2. Non-zero vector perpendicular to the tangent plane at a point M (x 0, y 0) surfaces z = f(x, y), is called normal to the surface at that point.

As a normal to the surface under consideration, it is convenient to take the vector - n = { , ,-1}.

Partial derivatives of a function in the event that they exist not at one point, but on a certain set, are functions defined on this set. These functions may be continuous and in some cases may also have partial derivatives at various points in the domain.

The partial derivatives of these functions are called second-order partial derivatives or second partial derivatives.

Second order partial derivatives are divided into two groups:

second partial derivatives of with respect to the variable;

· mixed partial derivatives of with respect to the variables and.

With subsequent differentiation, third-order partial derivatives can be determined, and so on. Higher-order partial derivatives are defined and written by analogous reasoning.

Theorem. If all partial derivatives included in the calculation, considered as functions of their independent variables, are continuous, then the result of partial differentiation does not depend on the sequence of differentiation.

Often there is a need to solve an inverse problem, which consists in determining whether the total differential of a function is an expression of the form where continuous functions with continuous derivatives of the first order.

The necessary condition for the total differential can be formulated as a theorem, which we accept without proof.

Theorem. In order for a differential expression to be in a domain the total differential of a function defined and differentiable in this domain, it is necessary that the condition for any pair of independent variables u be identically satisfied in this domain.

The task of calculating the total second-order differential of a function can be solved as follows. If the expression of the total differential is also differentiable, then the second total differential (or second-order total differential) can be considered the expression obtained as a result of applying the differentiation operation to the first total differential, i.e. . The analytical expression for the second total differential is:

Taking into account the fact that mixed derivatives do not depend on the order of differentiation, the formula can be grouped and represented as a quadratic form:

The quadratic form matrix is:

Let the superposition of functions defined in and

Certain in. Wherein. Then, if and have continuous partial derivatives up to the second order at the points and, then there is a second total differential of the compound function of the following form:

As you can see, the second total differential does not have the shape invariance property. The expression of the second differential of a complex function includes terms of the form that are absent in the formula of the second differential of a simple function.

The construction of partial derivatives of a function of higher orders can be continued by performing successive differentiation of this function:

Where indices take values ​​from to, i.e. the order derivative is considered as the first order partial derivative of the order derivative. Similarly, we can introduce the concept of the total differential of the order of a function, as the first order total differential of the order differential: .

In the case of a simple function of two variables, the formula for calculating the total differential of the order of a function is

The use of the differentiation operator makes it possible to obtain a compact and easy-to-remember notation for calculating the total differential of the order of a function, similar to Newton's binomial formula. In the two-dimensional case, it has the form

The concept of a function of two variables

Value z called function of two independent variables x and y, if each pair of admissible values ​​of these quantities, according to a certain law, corresponds to one well-defined value of the quantity z. Independent variables x and y called arguments functions.

Such a functional dependence is analytically denoted

Z = f (x, y),(1)

Values ​​of the x and y arguments that correspond to the actual values ​​of the function z, considered admissible, and the set of all admissible pairs of x and y values ​​is called domain of definition functions of two variables.

For a function of several variables, in contrast to a function of one variable, the concepts of its partial increments for each of the arguments and the concept full increment.

Partial increment Δ x z of the function z=f (x,y) by argument x is the increment that this function receives if its argument x is incremented Δx with the same y:

Δxz = f (x + Δx, y) -f (x, y), (2)

The partial increment Δ y z of the function z= f (x, y) with respect to the y argument is the increment that this function receives if its argument y receives an increment Δy with x unchanged:

Δy z= f (x, y + Δy) – f (x, y) , (3)

Full increment Δz functions z= f (x, y) by arguments x and y is called the increment that a function receives if both of its arguments are incremented:

Δz= f (x+Δx, y+Δy) – f (x, y) , (4)

For sufficiently small increments Δx and Δy function arguments

there is an approximate equality:

∆z ∆xz + ∆yz , (5)

and it is the more accurate, the less Δx and Δy.

Partial derivatives of functions of two variables

The partial derivative of the function z=f (x, y) with respect to the x argument at the point (x, y) is called the limit of the partial increment ratio ∆xz this function to the corresponding increment Δx argument x when striving Δx to 0 and provided that this limit exists:

, (6)

The derivative of the function is defined similarly z=f (x, y) by argument y:

In addition to the indicated notation, partial derivatives of functions are also denoted by , z΄ x , f΄ x (x, y); , z΄ y , f΄ y (x, y).

The main meaning of the partial derivative is as follows: the partial derivative of a function of several variables with respect to any of its arguments characterizes the rate of change of this function when this argument changes.

When calculating the partial derivative of a function of several variables with respect to any argument, all other arguments of this function are considered constant.

Example1. Find Partial Derivatives of Functions

f (x, y)= x 2 + y 3

Solution. When finding the partial derivative of this function with respect to the argument x, the argument y is considered a constant value:

;

When finding the partial derivative with respect to the argument y, the argument x is considered a constant value:

.

Partial and total differentials of a function of several variables

The partial differential of a function of several variables with respect to which-either from its arguments is the product of the partial derivative of this function with respect to the given argument and the differential of this argument:

dxz= ,(7)

dyz= (8)

Here d x z and d y z-partial differentials of a function z= f (x, y) by arguments x and y. Wherein

dx= ∆x; dy=Δy, (9)

full differential A function of several variables is called the sum of its partial differentials:

dz= d x z + d y z, (10)

Example 2 Find the partial and total differentials of the function f (x, y)= x 2 + y 3 .

Since the partial derivatives of this function are found in Example 1, we get

dxz= 2xdx; d y z= 3y 2 dy;

dz= 2xdx + 3y 2dy

The partial differential of a function of several variables with respect to each of its arguments is the principal part of the corresponding partial increment of the function.

As a result, one can write:

∆xz dxz, ∆yz d yz, (11)

The analytical meaning of the total differential is that the total differential of a function of several variables is the main part of the total increment of this function.

Thus, there is an approximate equality

∆zdz, (12)

The use of formula (12) is based on the use of the total differential in approximate calculations.

Imagine an increment Δz as

f (x + Δx; y + Δy) – f (x, y)

and the total differential in the form

Then we get:

f (x + Δx, y + Δy) – f (x, y) ,

, (13)

3. The purpose of the students in the lesson:

The student must know:

1. Definition of a function of two variables.

2. The concept of partial and total increment of a function of two variables.

3. Determination of the partial derivative of a function of several variables.

4. The physical meaning of the partial derivative of a function of several variables with respect to any of its arguments.

5. Determination of the partial differential of a function of several variables.

6. Determination of the total differential of a function of several variables.

7. Analytical meaning of the total differential.

The student must be able to:

1. Find private and total increments of a function of two variables.

2. Calculate partial derivatives of a function of several variables.

3. Find partial and total differentials of a function of several variables.

4. Apply the total differential of a function of several variables in approximate calculations.

Theoretical part:

1. The concept of a function of several variables.

2. Function of two variables. Partial and total increment of a function of two variables.

3. Partial derivative of a function of several variables.

4. Partial differentials of a function of several variables.

5. Total differential of a function of several variables.

6. Application of the total differential of a function of several variables in approximate calculations.

Practical part:

1.Find partial derivatives of functions:

1) ; 4) ;

2) z \u003d e xy + 2 x; 5) z= 2tg xx y;

3) z \u003d x 2 sin 2 y; 6) .

4. Define the partial derivative of a function with respect to a given argument.

5. What is called the partial and total differential of a function of two variables? How are they related?

6. List of questions to check the final level of knowledge:

1. In the general case of an arbitrary function of several variables, is its total increment equal to the sum of all partial increments?

2. What is the main meaning of the partial derivative of a function of several variables with respect to any of its arguments?

3. What is the analytical meaning of the total differential?

7. Timeline of the lesson:

1. Organizational moment - 5 minutes.

2. Analysis of the topic - 20 min.

3. Solving examples and problems - 40 min.

4. Current control of knowledge -30 min.

5. Summing up the lesson - 5 min.

8. List of educational literature for the lesson:

1. Morozov Yu.V. Fundamentals of higher mathematics and statistics. M., "Medicine", 2004, §§ 4.1–4.5.

2. Pavlushkov I.V. et al. Fundamentals of higher mathematics and mathematical statistics. M., "GEOTAR-Media", 2006, § 3.3.