Full differential and its applications. Chapter iv - functions of two variables. Derivative of a complex function. total derivative

Let Z= f(x;y) be defined in some neighborhood of the point M(x;y) total increment ∆Z=f(x+∆x,y+∆y)-f(x,y). Z= f(x;y) is called differentiable in M(x;y) if its total increment can be represented as: ∆Z=A∆x+B∆y+α∆x+β∆y, where α= α (∆х,∆у)→0 and β= β(∆х,∆у)→0 at ∆х→0, ∆у→0. The sum of the first two terms is the main part of the function increment. Main part of function increment, linear with respect to ∆x and ∆y, is called the total differential of the function and is denoted by the symbol dZ=A∆x+B∆y. The expressions A∆x and B∆y are called partial differentials. For independent variables x and y, we assume ∆x=dx, ∆y=dy. Therefore dZ=Adx+Bdy.

Theorem 1.(a necessary condition for differentiating a function). If Z = f(x; y) is differentiable at the point M(x; y), then it is continuous at this point and has partial derivatives at it , and =A; =V.

Thus, we can write dZ= dx+ dy or dZ=d x Z+ d y Z.

Theorem 2. If Z= f(x; y) has continuous partial derivatives Z′ x and Z′ y at the point M (x; y), then it is differentiable at this point and its total differential is expressed by the formula written higher.

For the function Z = f(x; y) to be differentiable at a point, it is necessary that it has partial derivatives at it and it is sufficient that it has continuous partial derivatives at the point.

The arithmetic properties of the rule for calculating the differentials of a function of one variable are also preserved in the case of differentials of a function of two or more variables.

1. Higher order differentials

The total differential is called the first order differential. Let Z= f(x; y) have continuous partial derivatives of the second order. The second-order differential in this case is determined by the formula
. Find it d 2 Z= d( dx+ dy)= ( dx+ dy) x ′ dx+( dx+ dy) y′ dу=( dx+ dy)dx+( dx+ dy)dу, hence d 2 Z= dx 2 +2 dxdy+ dy 2 . Symbolically, this can be written as follows: d 2 Z=(
) 2 Z. Similarly, one can obtain the formula

d 3 Z= d (d 2 Z)==(
) 3 Z, and for d n Z=(
) n Z. All these relations are valid only if the variables x and y of the function Z= f(x; y) are independent.

1. Derivative of a complex function. total derivative

Let Z= f(x;y) be a function of two variables x and y, each of which is a function of the independent variable t x=x(t),y=y(t). In this case, Z= f(x(t); y(t)) is a complex function of one independent variable t, and the variables x and y are intermediate variables.

Theorem. If Z= f(x; y) is differentiable at the point M(x, y) and x=x(t),y=y(t) are differentiable functions of the independent variable t, then the derivative of the composite function Z(t)= f( x(t);y(t)) is calculated by the formula
.

Proof. Let us increment ∆t to the independent t. Then x=x(t) and y=y(t) will receive increments ∆x and ∆y respectively. They, in turn, will cause the increment ∆Z of the Z function. Since Z \u003d f (x; y) is differentiable in M ​​(x, y) by the condition, then its total increment is equal to ∆Z \u003d
, where α→0 β →0 for ∆х→0 and ∆у→0. Divide ∆Z by ∆t and pass to the limit ∆t→0, then ∆x→0 and ∆y→0 due to the continuity of the functions x=x(t); y=y(t) we get:, i.e., Ch.t.d.

8. Invariance of the form of the total differential

Using the rules of differentiation of a complex function, one can show that the total differential has the property of invariance, i.e. retains the same form whether the arguments are independent variables or functions of independent variables.

Let Z= f(x; y), where x, y are independent variables, then the total differential (1st order) has the form dZ=

Consider a complex function Z= f(x; y), where x=x(u, ),y=y(u, ), i.e. function

informal description

Definitions

For functions

Differential of a smooth real-valued function f determined on M (M- a domain in or a smooth manifold) is a 1-form and is usually denoted df and is determined by the relation

where denotes the derivative f in the direction of the vector X in the tangent bundle M .

For displays

The differential of a smooth mapping from a smooth manifold to a manifold is a mapping between their tangent bundles , , such that for any smooth function we have

where Xf denotes the derivative f towards X. (On the left side of the equality, the derivative is taken in N functions g on dF(X) in the right - in M functions by X ).

This notion naturally generalizes the differential of a function.

Related definitions

Properties

Examples

Story

Term Differential(from lat. differentia- difference, difference) introduced by Leibniz. Initially, dx used to denote "infinitesimal" - a quantity that is less than any finite quantity and yet is not equal to zero. This view turned out to be inconvenient in most branches of mathematics (with the exception of non-standard analysis).

see also

Wikimedia Foundation. 2010 .

• Full Pe (studio)
• full duplex

See what "Full Differential" is in other dictionaries:

total differential- - [L.G. Sumenko. English Russian Dictionary of Information Technologies. M .: GP TsNIIS, 2003.] Topics information technology in general EN ordinary differentialtotal differential ... Technical Translator's Handbook

TOTAL DIFFERENTIAL- functions of n variables at a point is the same as the differential of a function at that point. The term P. d. is used to contrast it with the term partial differential. The concept of P. d. functions of n variables is generalized to the case of displaying open ... Mathematical Encyclopedia

Full differential- functions f (x, y, z, ...) of several independent variables expression in the case when it differs from the full increment (See Total increment) Δf = f (x + Δx, y + Δy, z + Δz,… ) f (x, y, z, …) on… … Great Soviet Encyclopedia

DIFFERENTIAL- (lat., from differe to distinguish). The limit of an infinitely small difference between a function of a variable that has received an infinitesimal increment and the original function of the same variable (mat. term.). Dictionary of foreign words included in the Russian ... ... Dictionary of foreign words of the Russian language

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Consider a function of two variables z=f(x, y) and its total increment at the point M 0 (x 0 , y 0)

Δ z \u003d f (x 0 + Δ x, y 0 + Δ y) - f (x 0, y 0).

Definition. If there are numbers P and Q such that the total increment can be represented as

Δz = PΔx + QΔy + ε Δρ,

where and ε→ 0 at Δρ→ 0 , then the expression PΔx + QΔy is called the total differential of the function z=f(x,y) at the point M0 (x0,y0).

In this case, the full increment of the function consists of two parts: the first part PΔx + QΔy is linear with respect to Δx and Δy, the second is an infinitesimal higher order in comparison with .

Total differential of a function z=f(x,y) denoted by dz, i.e

dz = PΔx+QΔy.

A function that has a total differential at a given point is called differentiable at that point.

Theorem. If a u=f(M) differentiable at a point M0, then it is continuous in it.

Comment. The continuity of a function of two variables does not imply its differentiability.

Example. continuous in (0,0) , but has no partial derivative - does not exist. Similarly, there is no partial derivative with respect to y. Therefore, the function is not differentiable.

Theorem [necessary condition for differentiability]. If a z=f(x,y) differentiable at a point M0, then it has partial derivatives with respect to x and y, and

f′ x (x 0 ,y 0) = P, f′ y (x 0 , y 0) = Q.

Comment. Differentiability does not follow from the existence of partial derivatives. Example:

We have , but the function is not continuous, hence it is not differentiable.

Theorem [sufficient condition for differentiability]. If the first partial derivatives of the functions z=f(x,y) are defined in some neighborhood of the point M0 (x0,y0) and continuous at the point M0, then the given function has a total differential at this point.

Comment. We have

Δ z \u003d f′ x (x 0, y 0)Δ x + f′ y (x 0, y 0)Δ y + ε Δρ,

where ε→ 0 at Δρ→ 0 . Hence,

f(x 0 +Δ x,y 0 +Δ y) - f(x 0 ,y 0) ≈ f′ x (x 0 ,y 0)Δ x + f′ y (x 0 ,y 0)Δ y

f(x 0 +Δ x, y 0 +Δ y) ≈ f(x 0 ,y 0) + f′ x (x 0 , y 0)Δ x + f′ y (x 0 , y 0)Δ y.

This formula is used in approximate calculations.

At fixed Δx and Δy the total differential is a function of the variables x and y:

Let's put dx=Δx, dy=Δy and call these quantities differentials of independent variables.

Then we get the formula

that is, the total differential of a function is equal to the sum of the products of the first partial derivatives and the corresponding differentials of the arguments.

The total differential of a function of three variables is defined and expressed similarly. If a u=f(x, y, z) and there are numbers P, Q, R such that

Δu = PΔx+QΔy+RΔz+εΔρ, ε→ 0 at δρ→ 0 ,

then the total differential is the expression

du = PΔx+QΔy+RΔz.

If the first partial derivatives of this function are continuous, then

where dx=Δx, dz=Δz, dz=Δz.

Definition. The total second-order differential of some function is the total differential of its total differential.

If a z=f(x,y), dz=z′ x dx+z′ y dy, then

Tangent plane and surface normal

Consider the surface S, given by the equation

z=f(x, y).

Let be f(x, y) has partial derivatives in some domain. Consider M 0 (x 0 , y 0).

- slope of the tangent at the point M0 to the section of the surface by the plane y=y0, that is, to the line z=f(x,y 0). The tangent to this line is:

z-z 0 \u003d f′ x (x 0, y 0) (x-x 0), y=y 0.

Similarly, a section by a plane x=x0 gives the equation

z-z 0 =f′ y (x 0 , y 0)(y-y 0), x=x 0.

The plane containing both of these lines has the equation

z-z 0 \u003d f′ x (x 0, y 0)(x-x 0)+f′ y (x 0, y 0)(y-y 0)

and is called the tangent plane to the surface S at the point P 0 (x 0 , y 0 , z 0).

Note that the tangent plane equation can be rewritten as

z-z 0 =df.

Thus, the geometric meaning of the total differential is: the differential at a point M0 for increment (x-x 0 , y-y 0) is the increment of the applicate point of the tangent plane to the surface z=f(x,y) at the point (x0, y0) for the same increments.

The tangent plane has a normal vector at the point (x0, y0, z0) - \vec(n)=(f′ x (x 0 , y 0), f′ y (x 0 , y 0), -1). A line passing through a point P0 and having a direction vector \vec(n), is called the normal to the surface z=f(x,y) at this point. Her equations are:

Differentiation of complex functions

Let a differentiable function be given z=F(v, w), whose arguments are differentiable functions of variables x and y:

v=v(x, y), w=w(x, y).

If at the same time the function

z=F(v(x, y), w(x, y))=\Phi(x, y)

makes sense, then it is called a complex function of x and y.

Theorem. Partial derivatives z′ x, z'y complex functions exist and are expressed by the formulas

If a v and w- differentiable functions of one variable t, i.e

v=v(t), w=w(t),

and the function makes sense

z=F(v(t), w(t))=f(t),

then its derivative is expressed by the formula

This derivative is called the total derivative.

If a differentiable function is given

u=F(ξ, η, ζ),

whose arguments ξ=ξ(t), η=η(t), ζ=ζ(t)- differentiable functions of a variable t and function

u=F(ξ(t), η(t), ζ(t))

IV. FUNCTIONS OF TWO VARIABLES

§ 10. Fundamentals of differentiating a function of two variables

Partial derivative from function to variable x is the limit

.

Partial derivative of a function
by variable y is the limit

.

Relevant designations: and , or and .

Derivative is the rate of change of the function with a small change in the variable x when the variable y constant. Obviously, is a new feature.

When searching believe that y is a number expressed by a letter (parameter). Then we get a function of one variable
, and the derivative of it is found according to the rules of differentiation of a function of one variable.

Same way is the rate of change of the function with a small change y and permanent x, while searching compose a function
and differentiate it as a function of one variable.

Example 1 Partial derivatives of a function:

Example 2 Let's find partial derivatives of a function
:

In the 1st case, a constant multiplier was taken out
, independent of x, and in the 2nd case - the multiplier , independent of y.

Example 3 For the function we find

Full differential
shows how about function will change if you increase x by the amount
and at the same time y- by the amount
(if
or
, then we are talking about a decrease x or y).

Example 4 Let's find the total differential of the function
in general and at the point
:

a)
- at
the derivative of the power function is obtained;

b)
- at
the derivative of the exponential function is obtained.

Thus, in general terms, or, if you take out the common factor,.

To find the total differential at a point by substituting its coordinates
and
, then.

Meaning of the result. Let it be necessary to find, for example, the value of the function
at the point
, or, which is the same, find the value
.

If we take a point
, then. When moving to a point N the change in the arguments was and (the difference between the old and new coordinates).

Total differential at a point M (not in N! )

is equal to the increment of the function when passing from the point
in
.

So . More accurately,
.

Example 5 Let's find total differentials for several functions in general form and at a specific point M:

a) let be
;
, then

Differential in general

at the point M will

b) let them be given
and
; then

General differential:

in) if given
and
, then

Simplify the numerators:

;
.

In the total differential, we take out the common factor:

let's substitute the coordinates of the point:

or
.

So to find
, we believe
, then, after which

and correspondingly .

Example 6 Using the total differential, we find the value of the function
at
(the angle is expressed in radians).

Pick a point as close as possible to
so that it can easily calculate the value
. This is the point
:
.

Partial derivatives in general:

,,

and at the point
will and
.

So near the point
the function changes in much the same way as the variable changes x. In our case .

The new value of the function.

A more accurate value almost coincides with the approximate one. The difference is due to the fact that
, not 1;

Answer: .

Example 7 Using the total differential, we find
.

Let's represent this number as the value of the function
at the point
. Wherein
and
, and for such arguments the function
easy to calculate:
.

So,
,
,
,
.

Then
prii.

For
partial derivatives

;
.

At the point M
and
, then

(the function grows 2 times faster than the 2nd argument).

Answer:(a more accurate value is
).

ChP1. Find partial derivatives for functions

3) a); b)
;

in)
; G)
;

State of emergency2. Find the total differentials of the functions at the given point:

2) a)
; b)
;

in)
; G)
;

ChP3. Find approximate values ​​using the total differential

1) a)
; b)
; in)
; G)
;

2) a)
; b)
; in)
; G)
;

3) a)
; b)
; in)
; G)
;

4) a) ; b) ; in) ; G) .

Extremum of a function of two variables

Dot M is called the minimum point of the function
, if you can specify an open area D(part of the plane xOy), in which the value
- the smallest of all. More strictly, M is the minimum point, if it exists D, what

a)
(the point is included in this area and does not belong to its boundary);

b) (at any other point in the same area, the value of the function is less than at the point of interest to us).

When replaced by the condition
we obtain the definition of the maximum point.

For example,
is the minimum point of the function, since at it, and at any other point
.

Scheme for searching for extremum points for a function

1) Find and , then - points
, where both derivatives are equal to 0;

2) find 2nd derivatives
, i.e. respectively
;

3) point coordinates
substitute into the 2nd derivatives. Let's get the numbers

4) if
, at the point
there is no extremum. If a
, then we look at the sign A:

if
, then
- minimum point,

if
, then
– maximum point;

5) if in
it turned out that
, other solution methods are needed that go beyond the scope of the manual (Taylor series expansion);

6) in the same way, the 3rd, 4th and 5th steps are performed for the remaining points.

Example 8 Let's find the extrema of the function .

1)

solve the system

(the equations are solved independently, and all combinations of coordinates are suitable);

2) find 2nd derivatives

;

;

Checking the point
, substituting
and
:

3) ;
;
;

4) , extremum in
no.

Checking the point
, substituting
and
:

3) ;
;
;

4) , extremum in
there is.

Insofar as
, then this extremum is the minimum. You can find its meaning.

Answer: minimum at
and
, equal to -50.

Example 9 We investigate the function for an extremum.

1) Find

solve the system

The 2nd equation has 3 roots: -1, 0 and 1, but the coordinates are dependent:

if
, then
,

if
, then
,

if
, then
.

We get 3 points: ;

2) take 2nd derivatives

;
;
;

check point
:

3)
;
;
;

4) , in
is an extremum, and since
, then this extremum is the minimum. Its meaning;

check point
:

3)
;
;
;

4) , extremum in
no.

It is easy to see that for the point
the results are the same as for
.

Answer: minimum equal to -2, with
and
, as well as at
and
.

Remark 1. If all signs are changed in the function record, the minimum points will become maximum points, and vice versa. In this case, the coordinates of the points will not change. So, from example 9 it follows that for we get a maximum equal to 2, with
and
, as well as at
and
.

If any number is added (or subtracted) to the function, only the value of the extremum will change, but not its type. Thus, the function will have a maximum at
and
, as well as at
and
, equal to 2+50=52.

ChP4.a, b. Find the value of the function at this point and determine the type of extremum:

a) a = 2; b= 3; b) a = 3; b= 2; in) a = 2; b= 5; G) a = 5; b = 4;

e) a = 6; b= 1; e) a = 1; b= 2; g) a = 0; b= 4; h) a = 3; b = 0.

ChP5. Find the extremum point of the function with the specified parameters a, b. Find the value of the function, determine the type of extremum:

a) a = 2; b= 3; b) a = 3; b= 2; in) a = 2; b= 5; G) a = 5; b = 4;

e) a = 6; b= 1; e) a = 1; b= 2; g) a = 0; b= 4; h) a = 3; b = 0.

Remark 2. Functions of two variables behave more complicated than functions of one variable. So, when solving problems for an extremum:

a) even continuous functions can have several maximum points and no minimum points (or vice versa);

b) all stationary points can be saddle points, from which the function grows when changing x and decreases when changing y(or vice versa). Thus, the function will not have a maximum or minimum.

Remark 3. The above scheme for studying the extremum assumes that the function is differentiable at the extremum points. However, this is not required. Yes, the function
at the point
has a maximum, but its derivatives at a given point go to infinity. Such cases are outside the scope of the grant.

ChP6. Examine the functions for an extremum and indicate the value of the extremum.