Function of several real variables

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 Template:Mvar variables is the subset of Template: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 Template:Tmath.

General definition

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

$f:X\to ℝ$

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

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

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

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

$\begin{array}{cc}& V:X\to ℝ\\ & X=\{(A,h)\in {ℝ}^2\mid A>0,h>0\}\\ & V(A,h)=\frac{1}{3}Ah\end{array}$

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

$\begin{array}{cc}& z:{ℝ}^2\to ℝ\\ & z(x,y)=ax+by\end{array}$

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

$\begin{array}{cc}& z:{ℝ}^p\to ℝ\\ & z(x_1,x_2,\dots ,x_p)=a_1{x}_1+a_2{x}_2+\cdots +a_p{x}_p\end{array}$

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

The Euclidean norm:

$f(𝒙)=\Vert 𝒙\Vert =\sqrt{x_1^2+\cdots +x_n^2}$

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

$g(𝒙)=\frac{1}{f(𝒙)}$

is defined only for Template:Math.

For a non-linear example function in two variables:

$\begin{array}{cc}& z:X\to ℝ\\ & X=\{(x,y)\in {ℝ}^2:x^2+y^2\le 8,x\ne 0,y\ne 0\}\\ & z(x,y)=\frac{1}{2xy}\sqrt{x^2+y^2}\end{array}$

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

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

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

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

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

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

Image

The image of a function Template:Math is the set of all values of Template:Mvar when the Template:Math-tuple Template:Math runs in the whole domain of Template: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 Template:Math is called a level set. It is the set of the solutions of the equation Template:Math.

Domain

The domain of a function of several real variables is a subset of Template:Math that is sometimes, but not always, explicitly defined. In fact, if one restricts the domain Template:Math of a function Template:Math to a subset Template:Math, one gets formally a different function, the restriction of Template:Math to Template:Math, which is denoted $f{|}_Y$. In practice, it is often (but not always) not harmful to identify Template:Math and $f{|}_Y$, and to omit the restrictor Template: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 Template:Math, it may be difficult to specify the domain of the function $g(𝒙)=1/f(𝒙).$ If Template:Math is a multivariate polynomial, (which has ${ℝ}^n$ as a domain), it is even difficult to test whether the domain of Template:Math is also ${ℝ}^n$. 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:

• For every real number Template:Math, the constant function $$(x_1,\dots ,x_n)\mapsto r$$ is everywhere defined.
• For every real number Template:Math and every function Template:Math, the function: $$rf:(x_1,\dots ,x_n)\mapsto rf(x_1,\dots ,x_n)$$ has the same domain as Template:Math (or is everywhere defined if Template:Math).
• If Template:Math and Template:Math are two functions of respective domains Template:Math and Template:Math such that Template:Math contains a nonempty open subset of Template:Math, then $$fg:(x_1,\dots ,x_n)\mapsto f(x_1,\dots ,x_n)g(x_1,\dots ,x_n)$$ and $$gf:(x_1,\dots ,x_n)\mapsto g(x_1,\dots ,x_n)f(x_1,\dots ,x_n)$$ are functions that have a domain containing Template:Math.

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

One may similarly define

$1/f:(x_1,\dots ,x_n)\mapsto 1/f(x_1,\dots ,x_n),$

which is a function only if the set of the points Template:Math in the domain of Template:Math such that Template:Math contains an open subset of Template: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 Template:Math is a point of the interior of the domain of the function Template:Math, we can fix the values of Template:Math to Template:Math respectively, to get a univariable function

$x\mapsto f(x,a_2,\dots ,a_n),$

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

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

$x\mapsto f(a_1+c_{1}x,a_2+c_{2}x,\dots ,a_n+c_{n}x),$

where the Template: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 Template:Math, which is an everywhere defined function of Template:Math real variables:

$d(𝒙,𝒚)=d(x_1,\dots ,x_n,y_1,\dots ,y_n)=\sqrt{(x_1-y_1{)}^2+\cdots +(x_n-y_n{)}^2}$

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

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

$f(x,y)=\frac{x^{2}y}{x^4+y^2}.$

The functions Template:Math and Template:Math are both constant and equal to zero, and are therefore continuous. The function Template:Math is not continuous at Template:Math, because, if Template:Math and Template:Math, we have Template:Math, even if Template: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 Template:Math are also continuous. In fact, we have

$f(x,\lambda x)=\frac{\lambda x}{x^2+{\lambda }^2}$

for Template:Math.

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

$L=\underset{𝒙\to 𝒂}{\lim }f(𝒙),$

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

$|f(𝒙)-L|<\epsilon$

for all Template:Math in the domain such that

$d(𝒙,𝒂)<\delta .$

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

$f(𝒂)=\underset{𝒙\to 𝒂}{\lim }f(𝒙).$

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

Symmetry

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

$f(\dots ,x_i,\dots ,x_j,\dots )=f(\dots ,x_j,\dots ,x_i,\dots )$

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

$f(x,y,z,t)=t^2-x^2-y^2-z^2$

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

Function composition

Suppose the functions

${\xi }_1={\xi }_1(x_1,x_2,\dots ,x_n),{\xi }_2={\xi }_2(x_1,x_2,\dots ,x_n),\dots {\xi }_m={\xi }_m(x_1,x_2,\dots ,x_n),$

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

$\mathit{\xi }:X\to \mathrm{\Xi }.$

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

$\begin{array}{cc}& \zeta :\mathrm{\Xi }\to ℝ,\\ & \zeta =\zeta ({\xi }_1,{\xi }_2,\dots ,{\xi }_m),\end{array}$

is a function composition defined on Template:Math, in other terms the mapping

$\begin{array}{cc}& \zeta :X\to ℝ,\\ & \zeta =\zeta ({\xi }_1,{\xi }_2,\dots ,{\xi }_m)=f(x_1,x_2,\dots ,x_n).\end{array}$

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

For example, the function

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

defined everywhere on Template:Math can be rewritten by introducing

$(\alpha ,\beta ,\gamma )=(\alpha (x,y),\beta (x,y),\gamma (x,y))=(xy,x-y,x+y)$

which is also everywhere defined in Template:Math to obtain

$f(x,y)=\zeta (\alpha (x,y),\beta (x,y),\gamma (x,y))=\zeta (\alpha ,\beta ,\gamma )=e^{\alpha }[\sin (3\beta )-\cos (2\gamma )].$

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

Partial derivatives can be defined with respect to each variable:

$\frac{\partial }{\partial x_1}f(x_1,x_2,\dots ,x_n),\frac{\partial }{\partial x_2}f(x_1,x_2,\dots x_n),\dots ,\frac{\partial }{\partial x_n}f(x_1,x_2,\dots ,x_n).$

Partial derivatives themselves are functions, each of which represents the rate of change of Template:Math parallel to one of the Template: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, Template:Math, its ordinary derivative Template:Math is geometrically the gradient of the tangent line to the curve Template: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:

$\frac{{\partial }^2}{\partial x_1^2}f(x_1,x_2,\dots ,x_n),\frac{{\partial }^2}{\partial x_1{x}_2}f(x_1,x_2,\dots x_n),\dots ,\frac{{\partial }^2}{\partial x_n^2}f(x_1,x_2,\dots ,x_n).$

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 Template:Math have the form:

$\frac{{\partial }^p}{\partial x_1^{p_1}\partial x_2^{p_2}\cdots \partial x_n^{p_n}}f(x_1,x_2,\dots ,x_n)\equiv \frac{{\partial }^{p_1}}{\partial x_1^{p_1}}\frac{{\partial }^{p_2}}{\partial x_2^{p_2}}\cdots \frac{{\partial }^{p_n}}{\partial x_n^{p_n}}f(x_1,x_2,\dots ,x_n)$

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

$\frac{{\partial }^0}{\partial x_1^0}f(x_1,x_2,\dots ,x_n)=f(x_1,x_2,\dots ,x_n),\dots ,\frac{{\partial }^0}{\partial x_n^0}f(x_1,x_2,\dots ,x_n)=f(x_1,x_2,\dots ,x_n).$

The number of possible partial derivatives increases with Template: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 Template:Math.

Multivariable differentiability

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

$f(𝒙)=f(𝒂)+𝑨(𝒂)\cdot (𝒙-𝒂)+\alpha (𝒙)|𝒙-𝒂|$

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

${\frac{\partial f(𝒙)}{\partial x_i}|}_{𝒙=𝒂}=A_i(𝒂)$

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

Assuming an Template: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:

$\mathrm{\nabla }f(𝒙)=(\frac{\partial }{\partial x_1},\frac{\partial }{\partial x_2},\dots ,\frac{\partial }{\partial x_n})f(𝒙)$

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 Template:Math (evaluated at Template:Math with a slight rearrangement gives:

$f(𝒙)-f(𝒂)=\mathrm{\nabla }f(𝒂)\cdot (𝒙-𝒂)+\alpha |𝒙-𝒂|$

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

$df={\frac{\partial f(𝒙)}{\partial x_1}|}_{𝒙=𝒂}d{x}_1+{\frac{\partial f(𝒙)}{\partial x_2}|}_{𝒙=𝒂}d{x}_2+\dots +{\frac{\partial f(𝒙)}{\partial x_n}|}_{𝒙=𝒂}d{x}_n=\mathrm{\nabla }f(𝒂)\cdot d𝒙$

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

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

$df=(\mathrm{\nabla }f)\cdot d𝒙=0$

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

In arbitrary curvilinear coordinate systems in Template: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 Template:Math in the domain:

${\frac{\partial }{\partial x_1}f(𝒙)|}_{𝒙=𝒂},{\frac{\partial }{\partial x_2}f(𝒙)|}_{𝒙=𝒂},\dots ,{\frac{\partial }{\partial x_n}f(𝒙)|}_{𝒙=𝒂}$

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

${\frac{{\partial }^p}{\partial x_1^{p_1}\partial x_2^{p_2}\cdots \partial x_n^{p_n}}f(𝒙)|}_{𝒙=𝒂}$

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

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

Multiple integration

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

$\underset{R_n}{\int }\cdots \underset{R_2}{\int }\underset{R_1}{\int }f(x_1,x_2,\dots ,x_n)d{x}_{1}d{x}_2\cdots d{x}_n\equiv \underset{R}{\int }f(𝒙)d^n𝒙$

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

$R_1\subseteq ℝ,R_2\subseteq ℝ,\dots ,R_n\subseteq ℝ,$

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

$R=R_1\times R_2\times \dots \times R_n,R\subseteq {ℝ}^n,$

an Template:Math-dimensional hypervolume. When evaluated, a definite integral is a real number if the integral converges in the region Template: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 Template:Math with respect to Template:Math has geometric interpretation as the area bounded by the curve Template:Math and the Template:Math-axis. Multiple integrals extend the dimensionality of this concept: assuming an Template:Math-dimensional analogue of a rectangular Cartesian coordinate system, the above definite integral has the geometric interpretation as the Template:Math-dimensional hypervolume bounded by Template:Math and the Template: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 Template:Math is some scalar density field and Template:Math are the position vector coordinates, i.e. some scalar quantity per unit n-dimensional hypervolume, then integrating over the region Template:Math gives the total amount of quantity in Template: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

$y_1=f_1(x_1,x_2,\dots ,x_n),y_2=f_2(x_1,x_2,\dots ,x_n),\dots ,y_m=f_m(x_1,x_2,\cdots x_n)$

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

$(y_1,y_2,\dots ,y_m)\leftrightarrow \left[\begin{array}{c}{f}_1(x_1,x_2,\dots ,x_n)\\ f_2(x_1,x_2,\cdots x_n)\\ ⋮\\ f_m(x_1,x_2,\dots ,x_n)\end{array}\right]\leftrightarrow \left[\begin{array}{cccc}{f}_1(x_1,x_2,\dots ,x_n)& f_2(x_1,x_2,\dots ,x_n)& \cdots & f_m(x_1,x_2,\dots ,x_n)\end{array}\right]$

all treated on the same footing as an Template:Math-component vector field, and use whichever form is convenient. All the above notations have a common compact notation Template: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 "Template:Math". Instead, the mapping is from the space Template:Math to the zero element in Template:Math (just the ordinary zero 0):

$\begin{array}{cc}& \varphi :{ℝ}^{n+1}\to \{0\}\\ & \varphi (x_1,x_2,\dots ,x_n,y)=0\end{array}$

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

$y=f(x_1,x_2,\dots ,x_n)$

then we can always define:

$\varphi (x_1,x_2,\dots ,x_n,y)=y-f(x_1,x_2,\dots ,x_n)=0$

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

For example, using interval notation, let

$\begin{array}{cc}& \varphi :X\to \{0\}\\ & \varphi (x,y,z)={\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2+{\left(\frac{z}{c}\right)}^2-1=0\\ & X=[-a,a]\times [-b,b]\times [-c,c]=\{(x,y,z)\in {ℝ}^3:-a\le x\le a,-b\le y\le b,-c\le z\le c\}.\end{array}$

Choosing a 3-dimensional (3D) Cartesian coordinate system, this function describes the surface of a 3D ellipsoid centered at the origin Template:Math with constant semi-major axes Template:Math, along the positive x, y and z axes respectively. In the case Template:Math, we have a sphere of radius Template: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 Template:Math, Template:Math or Template:Math; however it is much tidier to write it in an implicit form.

For a more sophisticated example:

$\begin{array}{cc}& \varphi :{ℝ}^4\to \{0\}\\ & \varphi (t,x,y,z)=Ctz{e}^{tx-yz}+A\sin (3\omega t)(x^{2}z-B{y}^6)=0\end{array}$

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

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

$\varphi (𝒂,b)=0;$

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

${\frac{\partial \varphi (𝒙,y)}{\partial y}|}_{(𝒙,y)=(𝒂,b)}\ne 0.$

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

$dy=\frac{\partial y}{\partial x_1}d{x}_1+\frac{\partial y}{\partial x_2}d{x}_2+\dots +\frac{\partial y}{\partial x_n}d{x}_n;$

$d\varphi =\frac{\partial \varphi }{\partial x_1}d{x}_1+\frac{\partial \varphi }{\partial x_2}d{x}_2+\dots +\frac{\partial \varphi }{\partial x_n}d{x}_n+\frac{\partial \varphi }{\partial y}dy.$

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

$\frac{\partial \varphi }{\partial x_i}+\frac{\partial \varphi }{\partial y}\frac{\partial y}{\partial x_i}=0$

for Template: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 Template:Math is such a complex valued function, it may be decomposed as

$f(x_1,\dots ,x_n)=g(x_1,\dots ,x_n)+ih(x_1,\dots ,x_n),$

where Template:Math and Template: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:

$z(x,y,\alpha ,a,q)=\frac{q}{2\pi }[\ln (x+iy-a{e}^{i\alpha })-\ln (x+iy+a{e}^{-i\alpha })]$

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 Template:Math of a mass distribution, a scalar field which depends on the spatial position coordinates (here Cartesian to exemplify), Template:Math, and time Template:Math:

$\rho =\rho (𝐫,t)=\rho (x,y,z,t)$

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 Template:Math that are each multivariable functions of spatial coordinates and time similarly:

$𝐯(𝐫,t)=𝐯(x,y,z,t)=[v_x(x,y,z,t),v_y(x,y,z,t),v_z(x,y,z,t)]$

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 Template:Math, temperature Template:Math, and volume Template:Math of a fluid, in general it has an implicit form:

$f(P,V,T)=0$

The simplest example is the ideal gas law:

$f(P,V,T)=PV-nRT=0$

where Template:Math is the number of moles, constant for a fixed amount of substance, and Template: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

$F(x,y,\dots )=\phi (x,y,\dots )+i\psi (x,y,\dots )$

is a complex valued function of the two spatial coordinates Template:Math and Template: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:

$Y_{\ell }^m=Y_{\ell }^m(\theta ,\varphi )$

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

$\mathrm{\Psi }=\mathrm{\Psi }(𝐫,t)=\mathrm{\Psi }(x,y,z,t),\mathrm{\Phi }=\mathrm{\Phi }(𝐩,t)=\mathrm{\Phi }(p_x,p_y,p_z,t)$

where each is related by a Fourier transform.

See also

• Real coordinate space
• Real analysis
• Complex analysis
• Function of several complex variables
• Multivariate interpolation
• Scalar fields

References

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