Total differentials and partial derivatives of higher orders. Full differential sign. Partial derivatives of a function of two variables. Concept and examples of solutions Partial derivatives and total differential

Let's consider changing a function when specifying an increment to only one of its arguments - x i, and let's call it .

Definition 1.7.Partial 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 proven for a function of one variable are valid 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 by 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 = xy,

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 select a point on the line of intersection of the plane and the surface M(x,y). If you give the argument at increment Δ at and consider 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 find 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 positive direction of the O axis u. Accordingly, the partial derivative is equal to the tangent of the angle with the O axis X tangent to the curve obtained as a result of sectioning the surface z = f(x,y) plane y = const.

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

Definition 2.2. If the increment of the function u = 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 principal linear part of the increment or total differential of the function in question.

Designations: 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 considered to be their arbitrary increments, therefore

Remark 1. So, the statement “the function is differentiable” is not equivalent to the statement “the function has partial derivatives” - for differentiability, the continuity of these derivatives at the point in question is also required.

4. Tangent plane and normal to the surface. Geometric meaning of 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 angular coefficients 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 create an equation for the plane passing through these lines. The tangent direction vectors 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 follows:

Where z 0 = .

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 a 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 a point M can be represented as:

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

In this case, the function differential f has the form:

which corresponds to Increment of applicates of a tangent plane to the graph of a function. This is the geometric meaning of the differential.

Definition 4.2. Nonzero vector perpendicular to the tangent plane at a point M (x 0 , y 0) surfaces z = f (x, y), called normal to the surface at this point.

It is convenient to take the vector -- n = { , ,-1}.

Concept of a function of two variables

Magnitude z called function of two independent variables x And y, if each pair of permissible values ​​of these quantities, according to a certain law, corresponds to one completely definite value of the quantity z. Independent Variables x And y called arguments functions.

This functional dependence is analytically denoted

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

The values ​​of the arguments x and y that correspond to the actual values ​​of the function z, are considered acceptable, and the set of all admissible pairs of values ​​x and y 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 private increments for each of the arguments and 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 constant y:

Δ x z = f (x + Δx, y) -f (x, y), (2)

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

Δ y z= f (x, y + Δy) – f (x, y) , (3)

Full increment Δz functions z=f(x,y) by argument x And y is the increment that a function receives if both of its arguments receive increments:

Δz= f (x+Δx, y+Δy) – f (x, y) , (4)

For sufficiently small increments Δx And Δy function arguments

there is an approximate equality:

Δz Δ x z + Δ y z , (5)

and the smaller it is, the more accurate it is Δx And Δy.

Partial derivatives of a function of two variables

Partial derivative of the function z=f (x, y) with respect to the argument x at the point (x, y) called the limit of the partial increment ratio Δ x z 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 determined similarly z=f(x,y) by argument y:

In addition to the indicated notation, partial derivative 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.

Example 1. Find partial derivatives of a function

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

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

;

When finding the partial derivative with respect to the argument y, we consider the argument x to be a constant value:

.

Partial and complete differentials of functions of several variables

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

d x z= ,(7)

d y z= (8)

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

dx=Δx; 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. Let's find the partial and complete differentials of the function f (x, y)= x 2 + y 3 .

Since the partial derivatives of this function were found in Example 1, we obtain

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

dz= 2xdx + 3y 2 dy

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

As a result, we can write:

Δ x z d x z, Δ y z d y z, (11)

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

Thus, there is an approximate equality

Δz dz, (12)

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

Let's imagine the increment Δz as

f (x + Δx; y + Δy) – f (x, y)

and the total differential is in the form

Then we get:

f (x + Δx, y + Δy) – f (x, y) ,

, (13)

3.The purpose of students’ activities in class:

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 the partial and total increment of a function of two variables.

2. Calculate partial derivatives of functions of several variables.

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

4. Use 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 functions of several variables.

5. Complete 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 the partial derivatives of the functions:

1) ; 4) ;

2) z= e xy+2 x; 5) z= 2tg xe y;

3) z= 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.Chronograph of the training session:

1. Organizational moment – ​​5 min.

2. Analysis of the topic – 20 min.

3. Solving examples and problems - 40 min.

4. Current knowledge control -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. and others. Fundamentals of higher mathematics and mathematical statistics. M., "GEOTAR-Media", 2006, § 3.3.

Partial derivative functions z = f(x, y by variable x The derivative of this function at a constant value of the variable y is called, it is denoted by or z" x.

Partial derivative functions z = f(x, y) by variable y is called the derivative with respect to y at a constant value of the variable y; it is designated or z" y.

The partial derivative of a function of several variables with respect to one variable is defined as the derivative of that function with respect to the corresponding variable, provided that the remaining variables are held constant.

Full differential function z = f(x, y) at some point M(X, y) is called the expression

,

Where and are calculated at the point M(x, y), and dx = , dy = y.

Example 1

Calculate the total differential of the function.

z = x 3 – 2x 2 y 2 + y 3 at point M(1; 2)

Solution:

1) Find partial derivatives:

2) Calculate the value of partial derivatives at point M(1; 2)

() M = 3 1 2 – 4 1 2 2 = -13

() M = - 4 1 2 2 + 3 2 2 = 4

3) dz = - 13dx + 4 dy

Questions for self-control:

1. What is called an antiderivative? List the properties of the antiderivative.

2. What is called an indefinite integral?

3. List the properties of the indefinite integral.

4. List the basic integration formulas.

5. What integration methods do you know?

6. What is the essence of the Newton–Leibniz formula?

7. Give the definition of a definite integral.

8. What is the essence of calculating a definite integral using the substitution method?

9. What is the essence of the method of calculating a definite integral by parts?

10. Which function is called a function of two variables? How is it designated?

11. Which function is called a function of three variables?

12. What set is called the domain of definition of a function?

13. Using what inequalities can you define a closed region D on a plane?

14. What is the partial derivative of the function z = f(x, y) with respect to the variable x? How is it designated?

15. What is the partial derivative of the function z = f(x, y) with respect to the variable y? How is it designated?

16. What expression is called the total differential of a function

Topic 1.2 Ordinary differential equations.

Problems leading to differential equations. Differential equations with separable variables. General and specific solutions. Homogeneous differential equations of the first order. Linear homogeneous equations of the second order with constant coefficients.

Practical lesson No. 7 “Finding general and particular solutions to differential equations with separable variables”*

Practical lesson No. 8 “Linear and homogeneous differential equations”

Practical lesson No. 9 “Solving 2nd order differential equations with constant coefficients”*

L4, chapter 15, pp. 243 – 256

Guidelines

Transcript

1 LECTURE N Total differential, partial derivatives and differentials of higher orders Total differential Partial differentials Partial derivatives of higher orders Differentials of higher orders 4 Derivatives of complex functions 4 Total differential Partial differentials If the function z=f(,) is differentiable, then its total differential dz is equal to dz= a +B () z z Noticing that A=, B =, we write formula () in the following form z z dz= + () Let’s extend the concept of a function differential to independent variables, putting the differentials of independent variables equal to their increments: d= ; d= After this, the formula for the total differential of the function will take the form z z dz= d + d () d + d Example Let =ln(+) Then dz= d + d = Similarly, if u=f(, n) is a differentiable function of n independent 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. From formulas (), (4) and (5) it follows that the total differential of a function is the sum of its partial differentials: dz=d z+d z Note that the total increment z of the function z=f(,), generally speaking, is not equal to the sum of partial increments. If at the point (,) the function z=f(,) is differentiable and the differential dz 0 at this point, then its total 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 infinitesimals of a higher order than the terms of the linear part Therefore at 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 in absolute value the increments of the arguments are,97 Example Calculate approximately arctg(),0

2 Solution Consider the function f(,)=arctg() Applying 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 applying 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, 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, with respect to, 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 partial derivative of the highest order, taken with respect to various variables, for example, ; called mixed partial derivative Example u= 4 z, then, u =4 z ; u = 4 z ; u z = 4 z; u = z ; u =6 4 z ; u zz = 4 ; u = z ; u = z ; u z = 4 z; u z =8 z; u z =6 4 z; u z =6 4 z Note that mixed derivatives taken with respect to the same variables, but in different orders, coincide. This property is not true for all functions, generally speaking, but it occurs in a wide class of functions. Theorem Assume that) the function f(,) is defined in the (open) domain D,) in this domain there exist the first derivatives f and f, as well as the second mixed derivatives f and f, and finally,) these last derivatives f and f, as functions of and, are continuous in some point (0, 0) of domain 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, positive, and are so small that D contains the entire rectangle [ 0, 0 +; 0, 0 +] Let us introduce an auxiliary function from: f (, 0 f (, 0) ϕ()=, which in the interval [ 0, 0 +] due to () has a derivative: f f ϕ (, 0 +) (, 0) ()= and therefore continuous With this function f (0 +, 0 f (0 +, 0) f (0, 0 f (0, 0) the expression W which is equal to W= can be rewritten in the form: ϕ (0 +) ϕ (0) W= Since for the function ϕ() in the interval [ 0, 0 +] all the conditions of the Lagrange theorem are satisfied, we can, using the finite increment formula, transform the expression W f 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 some 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 moving from one differential to the next (and d, d will be zeros) 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 for the function u there are continuous partial derivatives of all orders up to the nth inclusive, then the existence of the nth differential is guaranteed. But the expressions for them become more and more complex You can simplify the notation Let’s take the “letter u” out of brackets 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 parentheses is, formally, raised to a power according to the rules of algebra, then all the resulting terms are “multiplied” by u (which n is added to the numerators at), and only after This means that all symbols are returned to their meaning as derivatives and differentials u d) d u 4Derivatives of complex functions Let us have a function u=f(, z) defined in the domain D, and each of the variables, z, in turn, is a function of the variable t in some interval: =ϕ(t), =ψ(t), z=λ(t) Let, in addition, when t changes, the points (, z) do not go beyond the boundaries of the region D. Substituting the values, and z into the function u, we obtain complex function: u=f(ϕ(t), ψ(t), λ(t)) Assume that u and z have continuous partial derivatives u, u and u z and that t, t and z t exist Then we can prove existence of the derivative of a complex function and calculate it. Let's give the variable t some increment t, then z will receive increments, respectively, and z, and function u will receive increment u. Let's represent the increment of function u in the form: (this can be done, since we assumed the existence of continuous quotients derivatives u, u and u z) u=u +u +u z z+α +β +χ z, where α, β, χ 0 at, z 0 Divide both sides of the equality by t, we get u z z = u + u + uz + α + β + χ t t t t t t t 4

5 Let us now direct the increment of t to 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 a complex function really exists. If we use differential notation, then du d d dz () will have the form: = + + () dt dt dt z dt Let us now consider the case of dependence , z in several variables t: =ϕ(t, v), =ψ(t, v), z=χ(t, v) In addition to the existence and continuity of partial derivatives of the function f(, z), we assume here the existence of derivatives of functions, z in 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, a () needs to be rewritten in the form: = + + (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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To simplify the recording and presentation of the material, we will limit ourselves to the case of functions of two variables. Everything that follows is also true for functions of any number of variables.

Definition. Partial derivative functions z = f(x, y) by independent variable X called derivative

calculated at constant at.

The partial derivative with respect to a variable is determined similarly at.

For partial derivatives, the usual rules and formulas of differentiation are valid.

Definition. Product of the partial derivative and the increment of the argument X(y) is called partial differential by variable X(at) functions of two variables z = f(x, y) (symbol: ):

If under the differential of the independent variable dx(dy) understand increment X(at), That

For function z = f(x, y) let's find out the geometric meaning of its frequency derivatives and .

Consider the point, point P 0 (X 0 ,y 0 , z 0) on the surface z = f(x,at) and curve L, which is obtained by cutting the surface with a plane y = y 0 . This curve can be viewed as a graph of a function of one variable z = f(x, y) in the plane y = y 0 . If held at the point R 0 (X 0 , y 0 , z 0) tangent to the curve L, then, according to the geometric meaning of the derivative of a function of one variable , Where a – the angle formed by a tangent with the positive direction of the axis Oh.

Or: Let us similarly fix another variable, i.e. let's cross-section the surface z = f(x, y) plane x = x 0 . Then the function

z = f(x 0 , y) can be considered as a function of one variable at:

Where b– the angle formed by the tangent at the point M 0 (X 0 , y 0) with positive axis direction Oy(Fig. 1.2).

Rice. 1.2. Illustration of the geometric meaning of partial derivatives

Example 1.6. Given a function z = x 2 – 3xy – 4at 2 – x + 2y + 1. Find and .

Solution. Considering at as a constant, we get

Counting X constant, we find