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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Function of several real variables ： ウィキペディア英語版 Function of several real variables In mathematical analysis, and applications in geometry, applied mathematics, engineering, natural sciences, and economics, 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 several variables is the subset of for which the function is defined. As usual, the domain of a function of several real variables is supposed to contain an open subset of . ==General definition== |image2=Real function of two real variables.svg |caption2= |image3=Real function of three real variables.svg |caption3= |footer=Functions of variables, plotted as graphs in the space . The domains are the red -dimensional regions, the images are the purple -dimensional curves. }} A real-valued function of real variables is a function that takes as input real numbers, commonly represented by the variables , for producing another real number, the ''value'' of the function, commonly denoted . 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 of , the domain of the function, which is always supposed to contain an open subset of ℝ''n''. In other words, a real-valued function of real variables is a function :$f:\; X\; \backslash rightarrow\; \backslash mathbb$ such that its domain is a subset of that contains an open set. An element of being an -tuple (usually delimited by parentheses), the general notation for denoting functions would be . The common usage, much older than the general definition of functions between sets, it to not use double parentheses and to simply write . It is also common to abbreviate the -tuple by using a notation similar to that for vectors, like boldface , underline , or overarrow . This article will use bold. A simple example of a function in two variables could be: :$V\; :\; X\; \backslash rightarrow\; \backslash mathbb$ :$X\; =\; \backslash$ :$V(A,h)\; =\; \backslash fracA\; h$ which is the volume of a cone with base area and height 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: :$z\; :\; \backslash mathbb^2\; \backslash rightarrow\; \backslash mathbb$ :$z(x,y)\; =\; ax\; +\; by$ where and are real non-zero constants. Using the three-dimensional Cartesian coordinate system, where the xy plane is the domain and the z axis is the codomain , one can visualize the image to be a two-dimensional plane, with a slope of in the positive x direction and a slope of in the positive y direction. The function is well-defined at all points in . The previous example can be extended easily to higher dimensions: :$z\; :\; \backslash mathbb^p\; \backslash rightarrow\; \backslash mathbb$ :$z(x\_1,x\_2,\backslash ldots,\; x\_p)\; =\; a\_1\; x\_1\; +\; a\_2\; x\_2\; +\; \backslash cdots\; +\; a\_p\; x\_p$ for non-zero real constants , which describes a -dimensional hyperplane. The Euclidean norm: :$f(\backslash boldsymbol)=\backslash |\backslash boldsymbol\backslash |\; =\; \backslash sqrt$ is also a function of ''n'' variables which is everywhere defined, while :$g(\backslash boldsymbol)=\backslash frac$ is defined only for . For a non-linear example function in two variables: :$z\; :\; X\; \backslash rightarrow\; \backslash mathbb$ :$X\; =\; \backslash$ :$z(x,y)\; =\; \backslash frac\backslash sqrt$ which takes in all points in , a disk of radius "punctured" at the origin in the plane , and returns a point in . The function does not include the origin , if it did then would be ill-defined at that point. Using a 3d Cartesian coordinate system with the xy plane as the domain , and the z axis the codomain , the image can be visualized as a curved surface. The function can be evaluated at the point in : :$z(2,\backslash sqrt)\; =\; \backslash frac)^2\}\; =\; \backslash frac\; \backslash ,,$ However, the function couldn't be evaluated at, say :$(x,y)\; =\; (65,\backslash sqrt)\; \backslash ,\; \backslash Rightarrow\; \backslash ,\; x^2\; +\; y^2\; =\; (65)^2\; +\; (\backslash sqrt)^2\; >\; 8$ since these values of and do not satisfy the domain's rule. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Function of several real variables」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.