Definition of multivariate function
Let D be a non empty set of n-ary ordered arrays, and f be a certain Corresponding rules 。 [1] 
Figure 1 Multivariate function -- the image of binary function z=f (x, y) When n=1 Unary function , recorded as y=f (x), x ∈ D; When n=2 Bivariate function , z=f (x, y), (x, y) ∈ D, the image is shown in Figure 1. Functions of two or more variables are collectively referred to as multivariate functions.
Let D be a set of points in n-dimensional space, and f be a certain Correspondence rule 。 If for each point P , the variable z always has a uniquely determined value corresponding to it according to the corresponding rule f, then z is called the variable x one ,x two ,…,x n The n-ary function of. Recorded as , where , or z=f (P), P ∈ D. If the domain D of function f is Set of real numbers One of R subset , that is, it only depends on one independent variable Let's say that f is a function of one variable. If the domain D of function f is the Cartesian product of n R's R × R ×... × R= A subset of, that is, depends on n independent variables, that is, f is a function of n variables. When n ≥ 2, n-ary functions are generally called multivariate functions. [1]
The definition domain of binary function is usually a plane region enclosed by one or more smooth curves on the plane. The curve enclosed in the region is called Boundary of area The area including the boundary is called Closed region , otherwise called Open area 。 aggregate , which is called the definition field of a function, or D (f) or 。 [2] Correspondence rules (also called correspondence, correspondence rules, correspondence rules), f can be expressed in mathematical expressions (including analytic expressions), images, tables, etc.
about The corresponding y value is recorded as Is called when When, function The function value of. Set of all function values It is called the value field of the function, and is recorded as Z or Z (f). It is often said that the function y=f (x) is dependent variable With a independent variable The relationship between, that is, the value of the dependent variable only depends on one independent variable, is called a function of one variable. [1] But in many practical problems, it is often necessary to study the relationship between the dependent variable and several independent variables, that is, the value of the dependent variable depends on several independent variables.
For example, the market demand for a certain commodity is not only related to its market price, but also related to such factors as consumers' income and the price of other substitutes for this commodity, that is, there are more than one factor determining the demand for this commodity. To study this kind of problem comprehensively, we need to introduce the concept of multivariate function.
The Method of Thinking in Studying Multivariate Functions
The thinking method of studying the function of one variable is the basis of studying the function of multiple variables, especially the function of two variables. The thinking method of studying binary function is the basis of studying multivariate function.
Properties of multivariate function
Like a function of one variable, it has Define Fields 、 range , independent variable, dependent variable and other concepts and properties. [3] Similarities and Differences of Three Definitions
Three definitions of multivariate function are given here. Limits and derivatives are ordered array definitions, n-dimensional space definitions and Cartesian product definitions. It can be said that the first two are equivalent. The latter is more extensive.
The essence of multivariate function is a relationship, which is two aggregate A definite correspondence between. The elements of these two sets can be numbers; It can also be point, line, face or body; It can also be vectors, matrices, and so on. The result corresponding to one or more elements can be unique, that is, single valued. It can also be multiple elements, that is, multivalued. [1] The most common function, as well as the "function" mentioned in our middle school mathematics textbooks, is actually (full name) a single valued real variable function with one variable, unless otherwise specified.
set sth. up , , if for each point , a rule f has a unique u ∈ U corresponding to it: f: G → U, , then f is called an n-variable function, G is the definition domain, and U is the value domain. Basic elementary function And its image. Power function, exponential function, logarithmic function, trigonometric function and anti trigonometric function are called basic elementary functions. ① power function ： (μ ≠ 0, μ is any real number definition field: when μ is a positive integer, it is (- ∞,+∞); when μ is a negative integer, it is (- ∞, 0) ∨ (0,+∞), μ=α (integer), when α is odd, it is (- ∞,+∞), when α is even, it is (0,+∞), μ=p/q, p, q coprime, discussed as a composite function.
② exponential function ： (a>0, a ≠ 1), the definition domain is (- ∞,+∞), the value domain is (0,+∞), when a>1 is a strictly monotonically increasing function (that is, when x2>x1, y2>y1), the graphs of 0 and y=log (x) are symmetric about the y-axis. ③ Logarithmic function ： (a>0), a is called the bottom, the definition domain is (0,+∞), and the value domain is (- ∞,+∞). a> 1 is strictly monotonically increasing, 0 The base 10 logarithm is called the common logarithm, abbreviated as lgx. In science and technology, the logarithm with e as the base, that is, the natural logarithm, is commonly used as lnx. ⑥ Hyperbolic function : hyperbolic sine , hyperbolic cosine , hyperbolic tangent / , hyperbolic cotangent / 。 
