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Bivariate function

Mathematical noun

Function of two variables and Unary function Similarly, the notation f and f (x, y) have different meanings, but it is customary to use the notation "f (x, y), (x, y) ∈ D" or "z=f (x, y), (x, y) ∈ D" to represent the binary function f on D. The notation f representing the binary function can also be arbitrarily selected. For example, it can also be recorded as z=φ (x, y), z=z (x, y), etc^[1]

Chinese name: Bivariate function
Foreign name: function of two variables
expression: z=f（x，y），（x，y）∈D

independent variable: X and y
Define Fields: D
Value range: f（D）={z|z=f（x，y），（x，y）∈D}
Image: In the space rectangular coordinate system Oxyz curved surface

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▪ Total differentiation
▪ Geometric meaning

definition

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set up

Is a two-dimensional space

One of Nonempty subset , called mapping

Is defined in

A binary function on, usually written as

, where point set

Is called the Define Fields ，

be called independent variable ，

be called dependent variable .

In the above definition

A pair of values of Ordered real number group

）Value of corresponding dependent variable

also known as

At point

At function value , recorded as

, i.e

. Function value

The set formed by the whole of is called function

Of range , recorded as

, i.e

.^[1]

Basic concepts

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Interior point, exterior point, boundary point

One on a given plane point set

, for

For example, any point on the plane must be one of the following three points:

（1）

of interior point

If for point

, there is a

, make U δ （M zero ）⊂⊂⊂⊂⊂ E, that is to say, it exists in

The full and small open circle for the heart belongs to

, then

Is the inner point of E

（2）

Outer point

If for point

, there is a

, make U δ （M zero ）Ø E=Ø, that is, there is a

Fully small open circle and

If not, it is called

Point outside E

（3）

of Boundary point

If for point

, arbitrary make U δ （M zero ）Existing in

There is something wrong

The point is that

U δ （M zero ）∨ E ≠ Ø and U δ （M zero ）⊄ E, then called

Boundary point of^[2]

Accumulation point

set sth. up

, point set

, if for any given

, point P Decanter neighborhood

There are always

The point in the is called

yes

Gathering point of^[3]

Open set, closed set, boundary

If point set

The middle point is all

The inner point is called

Is an open set; If point set E contains all boundary points of E, then E is called a closed set

The set of all boundary points is called

The boundary of

.^[2]

Connected set

If any two points in set E can be completely

Middle broken line When connected, it is called

by Connected set .^[2]

Open area, closed area

Connected open sets are called regions or open regions

The set of points formed by an open region connecting its boundaries is called Closed region .^[4]

Bounded set, unbounded set

For planar point sets

, if there is a positive number

, so that E ⊂ U (O, r), where

Is the origin of coordinates, so it is called

by Bounded set , otherwise

by Unbounded set .^[4]

limit

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Let function of two variables

Of Define Fields by

，

yes

Of Accumulation point . If a constant exists

, for any given Positive number

, there are always positive numbers

, properly used

Both

If it is true, it is called constant

Is the function f (x, y) when

The limit of time, recorded as

, also recorded as

In order to distinguish from the limit of a function of one variable, we call the limit of a function of two variables a double limit^[5]

It must be noted that the existence of double limit means

Tending in any way

When,

all Infinite approach In A. Therefore, if

In a special way, such as along a fixed line or curve

Even if

We can not conclude that the limit of the function exists from the infinite approach to a certain value

Tend to

When,

If it tends to different values, then it can be concluded that the limit of this function does not exist

As for the limit operation of binary functions, there are similar algorithms to those of univariate functions^[6]

Continuity

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If the function

At point

Limit exists at and is

, i.e

, then called function

stay

Continuous at

If the function

In area

If every point within is continuous, it is called a function

stay

Internally continuous^[7]

All binary Elementary function It is continuous in its definition area. The so-called definition area refers to the area contained in the definition area or Closed region .^[8]

Binary on Bounded Closed Domain D continuous function , must be on D Bounded , and can obtain its Maximum and minimum value .

A bivariate continuous function on a bounded closed domain D must obtain any value between the maximum and minimum

The continuous function of two variables on the bounded closed domain D must be on D Uniform continuity .^[9]

set up

If for any given positive number

, there are always positive numbers

, so that for

Any two points on

, as long as

Both

, then

stay

Upper uniformly continuous^[9]

Differentiability

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Partial derivative

Let the function of two variables

Arguments for

keep Constant value

, at this time

Becomes an independent variable

Of Unary function If this unary function

stay

At wechat Business If exists, the derivative is called a function

At point

Place pair

Of Partial derivative (or partial derivative ）, recorded as

，

.^[10]

For example, the function z=x two +xy+y two On (x zero ，y zero ）Partial derivative z of x at x （x zero ，y zero ）It's a function of one variable z=x two +xy zero +y zero two At x zero Derivative at, i.e. z x （x zero ，y zero ）=2x zero +y zero .

If the function has a partial derivative at every point (x, y) in the region D

，

The two partial derivatives are also binary functions of x and y in D^[11]

Bivariate function

Two partial derivatives of

，

It is still a bivariate function about x and y. If these two partial derivatives are then evaluated for x or y, the second order partial derivative of function z=f (x, y) is obtained. Obviously, there are four second order partial derivatives of bivariate functions, which are

，

They are also indicated by the following symbols:

，

The second and third second-order partial derivatives above contain partial derivatives for different independent variables, which is called mixed partial derivatives^[12]

Re alignment of second order partial derivative

The third order partial derivative, the fourth order partial derivative