Английская Википедия:Function of several real variables

Материал из Онлайн справочника
Перейти к навигацииПерейти к поиску

Шаблон:Short description Шаблон:More footnotes Шаблон:Functions

In mathematical analysis and its applications, 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 Шаблон:Mvar variables is the subset of Шаблон:Tmath 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 Шаблон:Tmath.

General definition

Шаблон:Multiple image

A real-valued function of Шаблон:Math real variables is a function that takes as input Шаблон:Math real numbers, commonly represented by the variables Шаблон:Math, for producing another real number, the value of the function, commonly denoted Шаблон:Math. 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 Шаблон:Math of Шаблон:Math, the domain of the function, which is always supposed to contain an open subset of Шаблон:Math. In other words, a real-valued function of Шаблон:Math real variables is a function

<math>f: X \to \R </math>

such that its domain Шаблон:Math is a subset of Шаблон:Math that contains a nonempty open set.

An element of Шаблон:Math being an Шаблон:Math-tuple Шаблон:Math (usually delimited by parentheses), the general notation for denoting functions would be Шаблон:Math. The common usage, much older than the general definition of functions between sets, is to not use double parentheses and to simply write Шаблон:Math.

It is also common to abbreviate the Шаблон:Math-tuple Шаблон:Math by using a notation similar to that for vectors, like boldface Шаблон:Math, underline Шаблон:Math, or overarrow Шаблон:Math. This article will use bold.

A simple example of a function in two variables could be:

<math>\begin{align}

& V : X \to \R \\ & X = \left\{ (A,h) \in \R^2 \mid A>0, h> 0 \right\} \\ & V(A,h) = \frac{1}{3}A h \end{align}</math>

which is the volume Шаблон:Math of a cone with base area Шаблон:Math and height Шаблон:Math 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:

<math>\begin{align}

& z : \R^2 \to \R \\ & z(x,y) = ax + by \end{align}</math>

where Шаблон:Math and Шаблон:Math are real non-zero constants. Using the three-dimensional Cartesian coordinate system, where the xy plane is the domain Шаблон:Math and the z axis is the codomain Шаблон:Math, one can visualize the image to be a two-dimensional plane, with a slope of Шаблон:Math in the positive x direction and a slope of Шаблон:Math in the positive y direction. The function is well-defined at all points Шаблон:Math in Шаблон:Math. The previous example can be extended easily to higher dimensions:

<math>\begin{align}

& z : \R^p \to \R \\ & z(x_1,x_2,\ldots, x_p) = a_1 x_1 + a_2 x_2 + \cdots + a_p x_p \end{align}</math>

for Шаблон:Math non-zero real constants Шаблон:Math, which describes a Шаблон:Math-dimensional hyperplane.

The Euclidean norm:

<math>f(\boldsymbol{x})=\|\boldsymbol{x}\| = \sqrt{x_1^2 + \cdots + x_n^2}</math>

is also a function of n variables which is everywhere defined, while

<math>g(\boldsymbol{x})=\frac{1}{f(\boldsymbol{x})}</math>

is defined only for Шаблон:Math.

For a non-linear example function in two variables:

<math>\begin{align}

& z : X \to \R \\ & X = \left\{ (x,y) \in \R^2 \, : \, x^2 + y^2 \leq 8 \, , \, x \neq 0 \, , \, y \neq 0 \right\} \\ & z(x,y) = \frac{1}{2xy}\sqrt{x^2 + y^2} \end{align}</math>

which takes in all points in Шаблон:Math, a disk of radius Шаблон:Math "punctured" at the origin Шаблон:Math in the plane Шаблон:Math, and returns a point in Шаблон:Math. The function does not include the origin Шаблон:Math, if it did then Шаблон:Math would be ill-defined at that point. Using a 3d Cartesian coordinate system with the xy-plane as the domain Шаблон:Math, and the z axis the codomain Шаблон:Math, the image can be visualized as a curved surface.

The function can be evaluated at the point Шаблон:Math in Шаблон:Math:

<math>z\left(2,\sqrt{3}\right) = \frac{1}{2 \cdot 2 \cdot \sqrt{3}}\sqrt{\left(2\right)^2 + \left(\sqrt{3}\right)^2} = \frac{1}{4\sqrt{3}}\sqrt{7} \,, </math>

However, the function couldn't be evaluated at, say

<math>(x,y) = (65,\sqrt{10}) \, \Rightarrow \, x^2 + y^2 = (65)^2 + (\sqrt{10})^2 > 8 </math>

since these values of Шаблон:Math and Шаблон:Math do not satisfy the domain's rule.

Image

The image of a function Шаблон:Math is the set of all values of Шаблон:Mvar when the Шаблон:Math-tuple Шаблон:Math runs in the whole domain of Шаблон:Mvar. For a continuous (see below for a definition) real-valued function which has a connected domain, the image is either an interval or a single value. In the latter case, the function is a constant function.

The preimage of a given real number Шаблон:Math is called a level set. It is the set of the solutions of the equation Шаблон:Math.

Domain

Шаблон:Unreferenced section

The domain of a function of several real variables is a subset of Шаблон:Math that is sometimes, but not always, explicitly defined. In fact, if one restricts the domain Шаблон:Math of a function Шаблон:Math to a subset Шаблон:Math, one gets formally a different function, the restriction of Шаблон:Math to Шаблон:Math, which is denoted <math>f|_Y</math>. In practice, it is often (but not always) not harmful to identify Шаблон:Math and <math>f|_Y</math>, and to omit the restrictor Шаблон:Math.

Conversely, it is sometimes possible to enlarge naturally the domain of a given function, for example by continuity or by analytic continuation.

Moreover, many functions are defined in such a way that it is difficult to specify explicitly their domain. For example, given a function Шаблон:Math, it may be difficult to specify the domain of the function <math>g(\boldsymbol{x}) = 1/f(\boldsymbol{x}).</math> If Шаблон:Math is a multivariate polynomial, (which has <math>\R^n</math> as a domain), it is even difficult to test whether the domain of Шаблон:Math is also <math>\R^n</math>. This is equivalent to test whether a polynomial is always positive, and is the object of an active research area (see Positive polynomial).

Algebraic structure

The usual operations of arithmetic on the reals may be extended to real-valued functions of several real variables in the following way:

It follows that the functions of Шаблон:Math variables that are everywhere defined and the functions of Шаблон:Math variables that are defined in some neighbourhood of a given point both form commutative algebras over the reals (Шаблон:Math-algebras). This is a prototypical example of a function space.

One may similarly define

<math>1/f : (x_1,\ldots,x_n) \mapsto 1/f(x_1,\ldots,x_n),</math>

which is a function only if the set of the points Шаблон:Math in the domain of Шаблон:Math such that Шаблон:Math contains an open subset of Шаблон:Math. This constraint implies that the above two algebras are not fields.

Univariable functions associated with a multivariable function

One can easily obtain a function in one real variable by giving a constant value to all but one of the variables. For example, if Шаблон:Math is a point of the interior of the domain of the function Шаблон:Math, we can fix the values of Шаблон:Math to Шаблон:Math respectively, to get a univariable function

<math>x \mapsto f(x, a_2, \ldots, a_n),</math>

whose domain contains an interval centered at Шаблон:Math. This function may also be viewed as the restriction of the function Шаблон:Math to the line defined by the equations Шаблон:Math for Шаблон:Math.

Other univariable functions may be defined by restricting Шаблон:Math to any line passing through Шаблон:Math. These are the functions

<math>x \mapsto f(a_1+c_1 x, a_2+c_2 x, \ldots, a_n+c_n x),</math>

where the Шаблон:Math are real numbers that are not all zero.

In next section, we will show that, if the multivariable function is continuous, so are all these univariable functions, but the converse is not necessarily true.

Continuity and limit

Until the second part of 19th century, only continuous functions were considered by mathematicians. At that time, the notion of continuity was elaborated for the functions of one or several real variables a rather long time before the formal definition of a topological space and a continuous map between topological spaces. As continuous functions of several real variables are ubiquitous in mathematics, it is worth to define this notion without reference to the general notion of continuous maps between topological space.

For defining the continuity, it is useful to consider the distance function of Шаблон:Math, which is an everywhere defined function of Шаблон:Math real variables:

<math>d(\boldsymbol{x},\boldsymbol{y})=d(x_1, \ldots, x_n, y_1, \ldots, y_n)=\sqrt{(x_1-y_1)^2+\cdots +(x_n-y_n)^2}</math>

A function Шаблон:Math is continuous at a point Шаблон:Math which is interior to its domain, if, for every positive real number Шаблон:Math, there is a positive real number Шаблон:Math such that Шаблон:Math for all Шаблон:Math such that Шаблон:Math. In other words, Шаблон:Math may be chosen small enough for having the image by Шаблон:Math of the ball of radius Шаблон:Math centered at Шаблон:Math contained in the interval of length Шаблон:Math centered at Шаблон:Math. A function is continuous if it is continuous at every point of its domain.

If a function is continuous at Шаблон:Math, then all the univariate functions that are obtained by fixing all the variables Шаблон:Math except one at the value Шаблон:Math, are continuous at Шаблон:Math. The converse is false; this means that all these univariate functions may be continuous for a function that is not continuous at Шаблон:Math. For an example, consider the function Шаблон:Math such that Шаблон:Math, and is otherwise defined by

<math>f(x,y) = \frac{x^2y}{x^4+y^2}.</math>

The functions Шаблон:Math and Шаблон:Math are both constant and equal to zero, and are therefore continuous. The function Шаблон:Math is not continuous at Шаблон:Math, because, if Шаблон:Math and Шаблон:Math, we have Шаблон:Math, even if Шаблон:Math is very small. Although not continuous, this function has the further property that all the univariate functions obtained by restricting it to a line passing through Шаблон:Math are also continuous. In fact, we have

<math> f(x, \lambda x) =\frac{\lambda x}{x^2+\lambda^2}</math>

for Шаблон:Math.

The limit at a point of a real-valued function of several real variables is defined as follows.[1] Let Шаблон:Math be a point in topological closure of the domain Шаблон:Math of the function Шаблон:Math. The function, Шаблон:Math has a limit Шаблон:Math when Шаблон:Math tends toward Шаблон:Math, denoted

<math>L = \lim_{\boldsymbol{x} \to \boldsymbol{a}} f(\boldsymbol{x}), </math>

if the following condition is satisfied: For every positive real number Шаблон:Math, there is a positive real number Шаблон:Math such that

<math>|f(\boldsymbol{x}) - L| < \varepsilon </math>

for all Шаблон:Math in the domain such that

<math>d(\boldsymbol{x}, \boldsymbol{a})< \delta.</math>

If the limit exists, it is unique. If Шаблон:Math is in the interior of the domain, the limit exists if and only if the function is continuous at Шаблон:Math. In this case, we have

<math>f(\boldsymbol{a}) = \lim_{\boldsymbol{x} \to \boldsymbol{a}} f(\boldsymbol{x}). </math>

When Шаблон:Math is in the boundary of the domain of Шаблон:Math, and if Шаблон:Math has a limit at Шаблон:Math, the latter formula allows to "extend by continuity" the domain of Шаблон:Math to Шаблон:Math.

Symmetry

A symmetric function is a function Шаблон:Math that is unchanged when two variables Шаблон:Math and Шаблон:Math are interchanged:

<math>f(\ldots, x_i,\ldots,x_j,\ldots) = f(\ldots, x_j,\ldots,x_i,\ldots)</math>

where Шаблон:Math and Шаблон:Math are each one of Шаблон:Math. For example:

<math>f(x,y,z,t) = t^2 - x^2 - y^2 - z^2 </math>

is symmetric in Шаблон:Math since interchanging any pair of Шаблон:Math leaves Шаблон:Math unchanged, but is not symmetric in all of Шаблон:Math, since interchanging Шаблон:Math with Шаблон:Math or Шаблон:Math or Шаблон:Math gives a different function.

Function composition

Suppose the functions

<math>\xi_1 = \xi_1(x_1,x_2,\ldots,x_n), \quad \xi_2 = \xi_2(x_1,x_2,\ldots,x_n), \ldots \xi_m = \xi_m(x_1,x_2,\ldots,x_n),</math>

or more compactly Шаблон:Math, are all defined on a domain Шаблон:Math. As the Шаблон:Math-tuple Шаблон:Math varies in Шаблон:Math, a subset of Шаблон:Math, the Шаблон:Math-tuple Шаблон:Math varies in another region Шаблон:Math a subset of Шаблон:Math. To restate this:

<math>\boldsymbol{\xi} : X \to \Xi .</math>

Then, a function Шаблон:Math of the functions Шаблон:Math defined on Шаблон:Math,

<math>\begin{align}

& \zeta : \Xi \to \R, \\ & \zeta = \zeta(\xi_1,\xi_2,\ldots,\xi_m), \end{align}</math>

is a function composition defined on Шаблон:Math,[2] in other terms the mapping

<math>\begin{align}

& \zeta : X \to \R , \\ & \zeta = \zeta(\xi_1,\xi_2,\ldots,\xi_m) = f(x_1,x_2,\ldots,x_n). \end{align}</math>

Note the numbers Шаблон:Math and Шаблон:Math do not need to be equal.

For example, the function

<math>f(x,y) = e^{xy}[\sin 3(x-y) - \cos 2(x+y)]</math>

defined everywhere on Шаблон:Math can be rewritten by introducing

<math>(\alpha, \beta, \gamma ) = (\alpha(x,y), \beta(x,y) , \gamma(x,y) ) = ( xy , x-y, x+y )</math>

which is also everywhere defined in Шаблон:Math to obtain

<math>f(x,y) = \zeta(\alpha(x,y),\beta(x,y),\gamma(x,y)) = \zeta(\alpha,\beta,\gamma) = e^\alpha[\sin (3\beta) - \cos (2\gamma)] \,.</math>

Function composition can be used to simplify functions, which is useful for carrying out multiple integrals and solving partial differential equations.

Calculus

Elementary calculus is the calculus of real-valued functions of one real variable, and the principal ideas of differentiation and integration of such functions can be extended to functions of more than one real variable; this extension is multivariable calculus.

Partial derivatives

Шаблон:Main

Partial derivatives can be defined with respect to each variable:

<math>\frac{\partial}{\partial x_1} f(x_1, x_2, \ldots, x_n)\,,\quad \frac{\partial}{\partial x_2} f(x_1, x_2, \ldots x_n)\,,\ldots, \frac{\partial}{\partial x_n} f(x_1, x_2, \ldots, x_n). </math>

Partial derivatives themselves are functions, each of which represents the rate of change of Шаблон:Math parallel to one of the Шаблон:Math axes at all points in the domain (if the derivatives exist and are continuous—see also below). A first derivative is positive if the function increases along the direction of the relevant axis, negative if it decreases, and zero if there is no increase or decrease. Evaluating a partial derivative at a particular point in the domain gives the rate of change of the function at that point in the direction parallel to a particular axis, a real number.

For real-valued functions of a real variable, Шаблон:Math, its ordinary derivative Шаблон:Math is geometrically the gradient of the tangent line to the curve Шаблон:Math at all points in the domain. Partial derivatives extend this idea to tangent hyperplanes to a curve.

The second order partial derivatives can be calculated for every pair of variables:

<math>\frac{\partial^2}{\partial x^2_1} f(x_1, x_2, \ldots, x_n)\,,\quad \frac{\partial^2}{\partial x_1 x_2} f(x_1, x_2, \ldots x_n)\,,\ldots, \frac{\partial^2}{\partial x^2_n} f(x_1, x_2, \ldots, x_n) .</math>

Geometrically, they are related to the local curvature of the function's image at all points in the domain. At any point where the function is well-defined, the function could be increasing along some axes, and/or decreasing along other axes, and/or not increasing or decreasing at all along other axes.

This leads to a variety of possible stationary points: global or local maxima, global or local minima, and saddle points—the multidimensional analogue of inflection points for real functions of one real variable. The Hessian matrix is a matrix of all the second order partial derivatives, which are used to investigate the stationary points of the function, important for mathematical optimization.

In general, partial derivatives of higher order Шаблон:Math have the form:

<math>\frac{\partial^p}{\partial x_1^{p_1}\partial x_2^{p_2}\cdots\partial x_n^{p_n}} f(x_1, x_2, \ldots, x_n) \equiv \frac{\partial^{p_1}}{\partial x_1^{p_1}} \frac{\partial^{p_2}}{\partial x_2^{p_2}} \cdots \frac{\partial^{p_n}}{\partial x_n^{p_n}} f(x_1, x_2, \ldots, x_n)</math>

where Шаблон:Math are each integers between Шаблон:Math and Шаблон:Math such that Шаблон:Math, using the definitions of zeroth partial derivatives as identity operators:

<math>\frac{\partial^0}{\partial x_1^0}f(x_1, x_2, \ldots, x_n) = f(x_1, x_2, \ldots, x_n)\,,\quad \ldots,\, \frac{\partial^0}{\partial x_n^0}f(x_1, x_2, \ldots, x_n)=f(x_1, x_2, \ldots, x_n)\,. </math>

The number of possible partial derivatives increases with Шаблон:Math, although some mixed partial derivatives (those with respect to more than one variable) are superfluous, because of the symmetry of second order partial derivatives. This reduces the number of partial derivatives to calculate for some Шаблон:Math.

Multivariable differentiability

Шаблон:Main

A function Шаблон:Math is differentiable in a neighborhood of a point Шаблон:Math if there is an Шаблон:Math-tuple of numbers dependent on Шаблон:Math in general, Шаблон:Math, so that:[3]

<math>f(\boldsymbol{x}) = f(\boldsymbol{a}) + \boldsymbol{A}(\boldsymbol{a})\cdot(\boldsymbol{x}-\boldsymbol{a}) + \alpha(\boldsymbol x)|\boldsymbol{x}-\boldsymbol{a}|</math>

where <math>\alpha(\boldsymbol x) \to 0</math> as <math>|\boldsymbol{x}-\boldsymbol{a}| \to 0</math>. This means that if Шаблон:Math is differentiable at a point Шаблон:Math, then Шаблон:Math is continuous at Шаблон:Math, although the converse is not true - continuity in the domain does not imply differentiability in the domain. If Шаблон:Math is differentiable at Шаблон:Math then the first order partial derivatives exist at Шаблон:Math and:

<math>\left.\frac{\partial f(\boldsymbol{x})}{\partial x_i}\right|_{\boldsymbol{x} = \boldsymbol{a}} = A_i (\boldsymbol{a}) </math>

for Шаблон:Math, which can be found from the definitions of the individual partial derivatives, so the partial derivatives of Шаблон:Math exist.

Assuming an Шаблон:Math-dimensional analogue of a rectangular Cartesian coordinate system, these partial derivatives can be used to form a vectorial linear differential operator, called the gradient (also known as "nabla" or "del") in this coordinate system:

<math>\nabla f(\boldsymbol{x}) = \left(\frac{\partial}{\partial x_1}, \frac{\partial}{\partial x_2}, \ldots, \frac{\partial}{\partial x_n} \right) f(\boldsymbol{x}) </math>

used extensively in vector calculus, because it is useful for constructing other differential operators and compactly formulating theorems in vector calculus.

Then substituting the gradient Шаблон:Math (evaluated at Шаблон:Math with a slight rearrangement gives:

<math>f(\boldsymbol{x}) - f(\boldsymbol{a})= \nabla f(\boldsymbol{a})\cdot(\boldsymbol{x}-\boldsymbol{a}) + \alpha |\boldsymbol{x}-\boldsymbol{a}|</math>

where Шаблон:Math denotes the dot product. This equation represents the best linear approximation of the function Шаблон:Math at all points Шаблон:Math within a neighborhood of Шаблон:Math. For infinitesimal changes in Шаблон:Math and Шаблон:Math as Шаблон:Math:

<math>df = \left.\frac{\partial f(\boldsymbol{x})}{\partial x_1}\right|_{\boldsymbol{x}=\boldsymbol{a}}dx_1 +

\left.\frac{\partial f(\boldsymbol{x})}{\partial x_2}\right|_{\boldsymbol{x}=\boldsymbol{a}}dx_2 + \dots + \left.\frac{\partial f(\boldsymbol{x})}{\partial x_n}\right|_{\boldsymbol{x}=\boldsymbol{a}}dx_n = \nabla f(\boldsymbol{a}) \cdot d\boldsymbol{x}</math>

which is defined as the total differential, or simply differential, of Шаблон:Math, at Шаблон:Math. This expression corresponds to the total infinitesimal change of Шаблон:Math, by adding all the infinitesimal changes of Шаблон:Math in all the Шаблон:Math directions. Also, Шаблон:Math can be construed as a covector with basis vectors as the infinitesimals Шаблон:Math in each direction and partial derivatives of Шаблон:Math as the components.

Geometrically Шаблон:Math is perpendicular to the level sets of Шаблон:Math, given by Шаблон:Math which for some constant Шаблон:Math describes an Шаблон:Math-dimensional hypersurface. The differential of a constant is zero:

<math>df = (\nabla f) \cdot d \boldsymbol{x} = 0</math>

in which Шаблон:Math is an infinitesimal change in Шаблон:Math in the hypersurface Шаблон:Math, and since the dot product of Шаблон:Math and Шаблон:Math is zero, this means Шаблон:Math is perpendicular to Шаблон:Math.

In arbitrary curvilinear coordinate systems in Шаблон:Math dimensions, the explicit expression for the gradient would not be so simple - there would be scale factors in terms of the metric tensor for that coordinate system. For the above case used throughout this article, the metric is just the Kronecker delta and the scale factors are all 1.

Differentiability classes

If all first order partial derivatives evaluated at a point Шаблон:Math in the domain:

<math>\left.\frac{\partial}{\partial x_1} f(\boldsymbol{x})\right|_{\boldsymbol{x}=\boldsymbol{a}}\,,\quad

\left.\frac{\partial}{\partial x_2} f(\boldsymbol{x})\right|_{\boldsymbol{x}=\boldsymbol{a}}\,,\ldots, \left.\frac{\partial}{\partial x_n} f(\boldsymbol{x})\right|_{\boldsymbol{x}=\boldsymbol{a}} </math>

exist and are continuous for all Шаблон:Math in the domain, Шаблон:Math has differentiability class Шаблон:Math. In general, if all order Шаблон:Math partial derivatives evaluated at a point Шаблон:Math:

<math>\left.\frac{\partial^p}{\partial x_1^{p_1}\partial x_2^{p_2}\cdots\partial x_n^{p_n}} f(\boldsymbol{x})\right|_{\boldsymbol{x}=\boldsymbol{a}}</math>

exist and are continuous, where Шаблон:Math, and Шаблон:Math are as above, for all Шаблон:Math in the domain, then Шаблон:Math is differentiable to order Шаблон:Math throughout the domain and has differentiability class Шаблон:Math.

If Шаблон:Math is of differentiability class Шаблон:Math, Шаблон:Math has continuous partial derivatives of all order and is called smooth. If Шаблон:Math is an analytic function and equals its Taylor series about any point in the domain, the notation Шаблон:Math denotes this differentiability class.

Multiple integration

Шаблон:Main

Definite integration can be extended to multiple integration over the several real variables with the notation;

<math>\int_{R_n} \cdots \int_{R_2} \int_{R_1} f(x_1, x_2, \ldots, x_n) \, dx_1 dx_2\cdots dx_n \equiv \int_R f(\boldsymbol{x}) \, d^n\boldsymbol{x}</math>

where each region Шаблон:Math is a subset of or all of the real line:

<math>R_1 \subseteq \mathbb{R} \,, \quad R_2 \subseteq \mathbb{R} \,, \ldots , R_n \subseteq \mathbb{R}, </math>

and their Cartesian product gives the region to integrate over as a single set:

<math>R = R_1 \times R_2 \times \dots \times R_n \,,\quad R \subseteq \mathbb{R}^n \,,</math>

an Шаблон:Math-dimensional hypervolume. When evaluated, a definite integral is a real number if the integral converges in the region Шаблон:Math of integration (the result of a definite integral may diverge to infinity for a given region, in such cases the integral remains ill-defined). The variables are treated as "dummy" or "bound" variables which are substituted for numbers in the process of integration.

The integral of a real-valued function of a real variable Шаблон:Math with respect to Шаблон:Math has geometric interpretation as the area bounded by the curve Шаблон:Math and the Шаблон:Math-axis. Multiple integrals extend the dimensionality of this concept: assuming an Шаблон:Math-dimensional analogue of a rectangular Cartesian coordinate system, the above definite integral has the geometric interpretation as the Шаблон:Math-dimensional hypervolume bounded by Шаблон:Math and the Шаблон:Math axes, which may be positive, negative, or zero, depending on the function being integrated (if the integral is convergent).

While bounded hypervolume is a useful insight, the more important idea of definite integrals is that they represent total quantities within space. This has significance in applied mathematics and physics: if Шаблон:Math is some scalar density field and Шаблон:Math are the position vector coordinates, i.e. some scalar quantity per unit n-dimensional hypervolume, then integrating over the region Шаблон:Math gives the total amount of quantity in Шаблон:Math. The more formal notions of hypervolume is the subject of measure theory. Above we used the Lebesgue measure, see Lebesgue integration for more on this topic.

Theorems

With the definitions of multiple integration and partial derivatives, key theorems can be formulated, including the fundamental theorem of calculus in several real variables (namely Stokes' theorem), integration by parts in several real variables, the symmetry of higher partial derivatives and Taylor's theorem for multivariable functions. Evaluating a mixture of integrals and partial derivatives can be done by using theorem differentiation under the integral sign.

Vector calculus

One can collect a number of functions each of several real variables, say

<math>y_1 = f_1(x_1, x_2, \ldots, x_n)\,,\quad y_2 = f_2(x_1, x_2, \ldots, x_n)\,,\ldots, y_m = f_m(x_1, x_2, \cdots x_n) </math>

into an Шаблон:Math-tuple, or sometimes as a column vector or row vector, respectively:

<math>(y_1, y_2, \ldots, y_m) \leftrightarrow \begin{bmatrix} f_1(x_1, x_2, \ldots, x_n) \\ f_2(x_1, x_2, \cdots x_n) \\ \vdots \\ f_m(x_1, x_2, \ldots, x_n) \end{bmatrix} \leftrightarrow \begin{bmatrix} f_1(x_1, x_2, \ldots, x_n) & f_2(x_1, x_2, \ldots, x_n) & \cdots & f_m(x_1, x_2, \ldots, x_n) \end{bmatrix} </math>

all treated on the same footing as an Шаблон:Math-component vector field, and use whichever form is convenient. All the above notations have a common compact notation Шаблон:Math. The calculus of such vector fields is vector calculus. For more on the treatment of row vectors and column vectors of multivariable functions, see matrix calculus.

Implicit functions

A real-valued implicit function of several real variables is not written in the form "Шаблон:Math". Instead, the mapping is from the space Шаблон:Math to the zero element in Шаблон:Math (just the ordinary zero 0):

<math>\begin{align}

& \phi: \R^{n+1} \to \{0\} \\ & \phi(x_1, x_2, \ldots, x_n, y) = 0 \end{align}</math>

is an equation in all the variables. Implicit functions are a more general way to represent functions, since if:

<math>y=f(x_1, x_2, \ldots, x_n) </math>

then we can always define:

<math> \phi(x_1, x_2, \ldots, x_n, y) = y - f(x_1, x_2, \ldots, x_n) = 0 </math>

but the converse is not always possible, i.e. not all implicit functions have an explicit form.

For example, using interval notation, let

<math>\begin{align}

& \phi : X \to \{ 0 \} \\ & \phi(x,y,z) = \left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 + \left(\frac{z}{c}\right)^2 - 1 = 0 \\ & X = [-a,a] \times [-b,b] \times [-c,c] = \left\{ (x,y,z) \in \R^3 \,:\, -a\leq x\leq a, -b\leq y\leq b, -c\leq z\leq c \right\} . \end{align}</math>

Choosing a 3-dimensional (3D) Cartesian coordinate system, this function describes the surface of a 3D ellipsoid centered at the origin Шаблон:Math with constant semi-major axes Шаблон:Math, along the positive x, y and z axes respectively. In the case Шаблон:Math, we have a sphere of radius Шаблон:Math centered at the origin. Other conic section examples which can be described similarly include the hyperboloid and paraboloid, more generally so can any 2D surface in 3D Euclidean space. The above example can be solved for Шаблон:Math, Шаблон:Math or Шаблон:Math; however it is much tidier to write it in an implicit form.

For a more sophisticated example:

<math>\begin{align}

& \phi : \R^4 \to \{ 0 \} \\ & \phi(t,x,y,z) = C tz e^{tx-yz} + A \sin(3\omega t) \left(x^2z - B y^6\right) = 0 \end{align}</math>

for non-zero real constants Шаблон:Math, this function is well-defined for all Шаблон:Math, but it cannot be solved explicitly for these variables and written as "Шаблон:Math", "Шаблон:Math", etc.

The implicit function theorem of more than two real variables deals with the continuity and differentiability of the function, as follows.[4] Let Шаблон:Math be a continuous function with continuous first order partial derivatives, and let ϕ evaluated at a point Шаблон:Math be zero:

<math>\phi(\boldsymbol{a}, b) = 0;</math>

and let the first partial derivative of Шаблон:Math with respect to Шаблон:Math evaluated at Шаблон:Math be non-zero:

<math>\left.\frac{\partial \phi(\boldsymbol{x},y)}{\partial y}\right|_{(\boldsymbol{x},y) = (\boldsymbol{a},b)} \neq 0 .</math>

Then, there is an interval Шаблон:Math containing Шаблон:Math, and a region Шаблон:Math containing Шаблон:Math, such that for every Шаблон:Math in Шаблон:Math there is exactly one value of Шаблон:Math in Шаблон:Math satisfying Шаблон:Math, and Шаблон:Math is a continuous function of Шаблон:Math so that Шаблон:Math. The total differentials of the functions are:

<math>dy=\frac{\partial y}{\partial x_1}dx_1 + \frac{\partial y}{\partial x_2}dx_2 + \dots + \frac{\partial y}{\partial x_n}dx_n ;</math>
<math>d\phi=\frac{\partial \phi}{\partial x_1}dx_1 + \frac{\partial \phi}{\partial x_2}dx_2 + \dots + \frac{\partial \phi}{\partial x_n}dx_n + \frac{\partial \phi}{\partial y}dy .</math>

Substituting Шаблон:Math into the latter differential and equating coefficients of the differentials gives the first order partial derivatives of Шаблон:Math with respect to Шаблон:Math in terms of the derivatives of the original function, each as a solution of the linear equation

<math>\frac{\partial \phi}{\partial x_i} + \frac{\partial \phi}{\partial y}\frac{\partial y}{\partial x_i} = 0 </math>

for Шаблон:Math.

Complex-valued function of several real variables

A complex-valued function of several real variables may be defined by relaxing, in the definition of the real-valued functions, the restriction of the codomain to the real numbers, and allowing complex values.

If Шаблон:Math is such a complex valued function, it may be decomposed as

<math>f(x_1,\ldots, x_n)=g(x_1,\ldots, x_n)+ih(x_1,\ldots, x_n),</math>

where Шаблон:Math and Шаблон:Math are real-valued functions. In other words, the study of the complex valued functions reduces easily to the study of the pairs of real valued functions.

This reduction works for the general properties. However, for an explicitly given function, such as:

<math> z(x, y, \alpha, a, q) = \frac{q}{2\pi} \left[\ln\left(x+iy- ae^{i\alpha}\right) - \ln\left(x+iy + ae^{-i\alpha}\right)\right]</math>

the computation of the real and the imaginary part may be difficult.

Applications

Multivariable functions of real variables arise inevitably in engineering and physics, because observable physical quantities are real numbers (with associated units and dimensions), and any one physical quantity will generally depend on a number of other quantities.

Examples of real-valued functions of several real variables

Examples in continuum mechanics include the local mass density Шаблон:Math of a mass distribution, a scalar field which depends on the spatial position coordinates (here Cartesian to exemplify), Шаблон:Math, and time Шаблон:Math:

<math>\rho = \rho(\mathbf{r},t) = \rho(x,y,z,t)</math>

Similarly for electric charge density for electrically charged objects, and numerous other scalar potential fields.

Another example is the velocity field, a vector field, which has components of velocity Шаблон:Math that are each multivariable functions of spatial coordinates and time similarly:

<math>\mathbf{v} (\mathbf{r},t) = \mathbf{v}(x,y,z,t) = [v_x(x,y,z,t), v_y(x,y,z,t), v_z(x,y,z,t)]</math>

Similarly for other physical vector fields such as electric fields and magnetic fields, and vector potential fields.

Another important example is the equation of state in thermodynamics, an equation relating pressure Шаблон:Math, temperature Шаблон:Math, and volume Шаблон:Math of a fluid, in general it has an implicit form:

<math>f(P, V, T) = 0 </math>

The simplest example is the ideal gas law:

<math>f(P, V, T) = PV - nRT = 0 </math>

where Шаблон:Math is the number of moles, constant for a fixed amount of substance, and Шаблон:Math the gas constant. Much more complicated equations of state have been empirically derived, but they all have the above implicit form.

Real-valued functions of several real variables appear pervasively in economics. In the underpinnings of consumer theory, utility is expressed as a function of the amounts of various goods consumed, each amount being an argument of the utility function. The result of maximizing utility is a set of demand functions, each expressing the amount demanded of a particular good as a function of the prices of the various goods and of income or wealth. In producer theory, a firm is usually assumed to maximize profit as a function of the quantities of various goods produced and of the quantities of various factors of production employed. The result of the optimization is a set of demand functions for the various factors of production and a set of supply functions for the various products; each of these functions has as its arguments the prices of the goods and of the factors of production.

Examples of complex-valued functions of several real variables

Some "physical quantities" may be actually complex valued - such as complex impedance, complex permittivity, complex permeability, and complex refractive index. These are also functions of real variables, such as frequency or time, as well as temperature.

In two-dimensional fluid mechanics, specifically in the theory of the potential flows used to describe fluid motion in 2d, the complex potential

<math>F(x,y,\ldots) = \varphi(x,y,\ldots) + i\psi(x,y,\ldots) </math>

is a complex valued function of the two spatial coordinates Шаблон:Math and Шаблон:Math, and other real variables associated with the system. The real part is the velocity potential and the imaginary part is the stream function.

The spherical harmonics occur in physics and engineering as the solution to Laplace's equation, as well as the eigenfunctions of the z-component angular momentum operator, which are complex-valued functions of real-valued spherical polar angles:

<math>Y^m_\ell = Y^m_\ell(\theta,\phi) </math>

In quantum mechanics, the wavefunction is necessarily complex-valued, but is a function of real spatial coordinates (or momentum components), as well as time Шаблон:Math:

<math>\Psi = \Psi(\mathbf{r},t) = \Psi(x,y,z,t)\,,\quad \Phi = \Phi(\mathbf{p},t) = \Phi(p_x,p_y,p_z,t) </math>

where each is related by a Fourier transform.

See also

References

Шаблон:Reflist

Шаблон:Analysis-footer Шаблон:Authority control

Партнерские ресурсы
Криптовалюты
Магазины
Хостинг
Разное

  1. ↑ Шаблон:Cite book
  2. ↑ Шаблон:Cite book
  3. ↑ Шаблон:Cite book
  4. ↑ Шаблон:Cite book
Источник — http://wikihandbk.com/ruwiki/index.php?title=Английская_Википедия:Function_of_several_real_variables&oldid=14819634