In mathematics, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The "input" variables take real values, while the "output", also called the "value of the function", may be real or complex. However, the study of the complex-valued functions may be easily reduced to the study of the real-valued functions, by considering the real and imaginary parts of the complex function; therefore, unless explicitly specified, only real-valued functions will be considered in this article. The domain of a function of n variables is the subset of R n {\displaystyle \mathbb {R} ^{n}} for which the function is defined. As usual, the domain of a function of several real variables is supposed to contain a nonempty open subset of R n {\displaystyle \mathbb {R} ^{n}} .
General definition
A real-valued function of n real variables is a function that takes as input n real numbers, commonly represented by the variables x1, x2, …, xn, for producing another real number, the value of the function, commonly denoted f(x1, x2, …, xn). For simplicity, in this article a real-valued function of several real variables will be simply called a function. To avoid any ambiguity, the other types of functions that may occur will be explicitly specified. Some functions are defined for all real values of the variables (one says that they are everywhere defined), but some other functions are defined only if the value of the variable are taken in a subset X of Rn, the domain of the function, which is always supposed to contain an open subset of Rn. In other words, a real-valued function of n real variables is a function
f : X → R {\displaystyle f:X\to \mathbb {R} }
such that its domain X is a subset of Rn that contains a nonempty open set. An element of X being an n-tuple (x1, x2, …, xn) (usually delimited by parentheses), the general notation for denoting functions would be f((x1, x2, …, xn)). The common usage, much older than the general definition of functions between sets, is to not use double parentheses and to simply write f(x1, x2, …, xn). It is also common to abbreviate the n-tuple (x1, x2, …, xn) by using a notation similar to that for vectors, like boldface x, underline x, or overarrow x→. This article will use bold. A simple example of a function in two variables could be:
V : X → R X = { ( A , h ) ∈ R 2 ∣ A > 0 , h > 0 } V ( A , h ) = 1 3 A h {\displaystyle {\begin{aligned}&V:X\to \mathbb {R} \\&X=\left\{(A,h)\in \mathbb {R} ^{2}\mid A>0,h>0\right\}\\&V(A,h)={\frac {1}{3}}Ah\end{aligned}}}
which is the volume V of a cone with base area A and height h measured perpendicularly from the base. The domain restricts all variables to be positive since lengths and areas must be positive. For an example of a function in two variables:
z : R 2 → R z ( x , y ) = a x + b y {\displaystyle {\begin{aligned}&z:\mathbb {R} ^{2}\to \mathbb {R} \\&z(x,y)=ax+by\end{aligned}}}
where a and b are real non-zero constants. Using the three-dimensional Cartesian coordinate system, where the xy plane is the domain R2 and the z axis is the codomain R, one can visualize the image to be a two-dimensional plane, with a slope of a in the positive x direction and a slope of b in the positive y direction. The function is well-defined at all points (x, y) in R2. The previous example can be extended easily to higher dimensions:
… excerpt ends here. Continue reading the full article.
