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Inmathematical analysis and its applications, afunction of several real variables orreal multivariate function is afunction with more than oneargument, with all arguments beingreal variables. This concept extends the idea of afunction 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 orcomplex. However, thestudy of the complex-valued functions may be easily reduced to thestudy of the real-valued functions, by considering the real andimaginary parts of the complex function; therefore, unless explicitly specified, only real-valued functions will be considered in this article.

Thedomain of a function ofn variables is thesubset of⁠${\displaystyle {\mathbb{R}}^n}$⁠ for which the function is defined. As usual, the domain of a function of several real variables is supposed to contain a nonemptyopen subset of⁠${\displaystyle {\mathbb{R}}^n}$⁠.

n = 1

n = 2

n = 3

Functionsf(x₁,x₂, …,x_n) ofn variables, plotted as graphs in the spaceR^{n + 1}. The domains are the redn-dimensional regions, the images are the purplen-dimensional curves.

Areal-valued function ofn real variables is afunction that takes as inputnreal numbers, commonly represented by thevariablesx₁,x₂, …,x_n, for producing another real number, thevalue of the function, commonly denotedf(x₁,x₂, …,x_n). For simplicity, in this article a real-valued function of several real variables will be simply called afunction. 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 subsetX ofRⁿ, thedomain of the function, which is always supposed to contain anopen subset ofRⁿ. In other words, a real-valued function ofn real variables is a function

${\displaystyle f:X\to \mathbb{R}}$

such that its domainX is a subset ofRⁿ that contains a nonempty open set.

An element ofX being ann-tuple(x₁,x₂, …,x_n) (usually delimited by parentheses), the general notation for denoting functions would bef((x₁,x₂, …,x_n)). The common usage, much older than the general definition of functions between sets, is to not use double parentheses and to simply writef(x₁,x₂, …,x_n).

It is also common to abbreviate then-tuple(x₁,x₂, …,x_n) by using a notation similar to that forvectors, like boldfacex, underlinex, or overarrowx→. This article will use bold.

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

which is thevolumeV of acone with base areaA and heighth measured perpendicularly from the base. The domain restricts all variables to be positive sincelengths andareas must be positive.

For an example of a function in two variables:

wherea andb are real non-zero constants. Using thethree-dimensionalCartesian coordinate system, where thexy plane is the domainR² and the z axis is the codomainR, one can visualize the image to be a two-dimensional plane, with aslope ofa in the positive x direction and a slope ofb in the positive y direction. The function is well-defined at all points(x,y) inR². The previous example can be extended easily to higher dimensions:

forp non-zero real constantsa₁,a₂, …,a_p, which describes ap-dimensionalhyperplane.

TheEuclidean norm:

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

is also a function ofn variables which is everywhere defined, while

${\displaystyle g(\mathit{x})=\frac{1}{f(\mathit{x})}}$

is defined only forx ≠ (0, 0, …, 0).

For a non-linear example function in two variables:

which takes in all points inX, adisk of radius√8 "punctured" at the origin(x,y) = (0, 0) in the planeR², and returns a point inR. The function does not include the origin(x,y) = (0, 0), if it did thenf would be ill-defined at that point. Using a 3d Cartesian coordinate system with thexy-plane as the domainR², and the z axis the codomainR, the image can be visualized as a curved surface.

The function can be evaluated at the point(x,y) = (2,√3) inX:

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

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

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

since these values ofx andy do not satisfy the domain's rule.

Theimage of a functionf(x₁,x₂, …,x_n) is the set of all values off when then-tuple(x₁,x₂, …,x_n) runs in the whole domain off. For a continuous (see below for a definition) real-valued function which has a connected domain, the image is either aninterval or a single value. In the latter case, the function is aconstant function.

Thepreimage of a given real numberc is called alevel set. It is the set of the solutions of theequationf(x₁,x₂, …,x_n) =c.

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Thedomain of a function of several real variables is a subset ofRⁿ that is sometimes, but not always, explicitly defined. In fact, if one restricts the domainX of a functionf to a subsetY ⊂X, one gets formally a different function, therestriction off toY, which is denoted${\displaystyle f{|}_Y}$. In practice, it is often (but not always) not harmful to identifyf and${\displaystyle f{|}_Y}$, and to omit the restrictor|_Y.

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

Moreover, many functions are defined in such a way that it is difficult to specify explicitly their domain. For example, given a functionf, it may be difficult to specify the domain of the function${\displaystyle g(\mathit{x})=1/f(\mathit{x}).}$ Iff is amultivariate polynomial, (which has${\displaystyle {\mathbb{R}}^n}$ as a domain), it is even difficult to test whether the domain ofg is also${\displaystyle {\mathbb{R}}^n}$. This is equivalent to test whether a polynomial is always positive, and is the object of an active research area (seePositive polynomial).

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 numberr, theconstant function$${\displaystyle (x_1,\dots ,x_n)\mapsto r}$$ is everywhere defined.
• For every real numberr and every functionf, the function:$${\displaystyle rf:(x_1,\dots ,x_n)\mapsto rf(x_1,\dots ,x_n)}$$ has the same domain asf (or is everywhere defined ifr = 0).
• Iff andg are two functions of respective domainsX andY such thatX ∩Y contains a nonempty open subset ofRⁿ, then$${\displaystyle fg:(x_1,\dots ,x_n)\mapsto f(x_1,\dots ,x_n)g(x_1,\dots ,x_n)}$$ and$${\displaystyle gf:(x_1,\dots ,x_n)\mapsto g(x_1,\dots ,x_n)f(x_1,\dots ,x_n)}$$ are functions that have a domain containingX ∩Y.

It follows that the functions ofn variables that are everywhere defined and the functions ofn variables that are defined in someneighbourhood of a given point both formcommutative algebras over the reals (R-algebras). This is a prototypical example of afunction space.

One may similarly define

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

which is a function only if the set of the points(x₁, …,x_n) in the domain off such thatf(x₁, …,x_n) ≠ 0 contains an open subset ofRⁿ. This constraint implies that the above two algebras are notfields.

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(a₁, …,a_n) is a point of theinterior of the domain of the functionf, we can fix the values ofx₂, …,x_n toa₂, …,a_n respectively, to get a univariable function

${\displaystyle x\mapsto f(x,a_2,\dots ,a_n),}$

whose domain contains an interval centered ata₁. This function may also be viewed as therestriction of the functionf to the line defined by the equationsx_i =a_i fori = 2, …,n.

Other univariable functions may be defined by restrictingf to any line passing through(a₁, …,a_n). These are the functions

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

where thec_i 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.

Until the second part of 19th century, onlycontinuous 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 atopological space and acontinuous 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 thedistance function ofRⁿ, which is an everywhere defined function of2n real variables:

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

A functionf iscontinuous at a pointa = (a₁, …,a_n) which isinterior to its domain, if, for every positive real numberε, there is a positive real numberφ such that|f(x) −f(a)| <ε for allx such thatd(xa) <φ. In other words,φ may be chosen small enough for having the image byf of the ball of radiusφ centered ata contained in the interval of length2ε centered atf(a). A function is continuous if it is continuous at every point of its domain.

If a function is continuous atf(a), then all the univariate functions that are obtained by fixing all the variablesx_i except one at the valuea_i, are continuous atf(a). The converse is false; this means that all these univariate functions may be continuous for a function that is not continuous atf(a). For an example, consider the functionf such thatf(0, 0) = 0, and is otherwise defined by

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

The functionsx ↦f(x, 0) andy ↦f(0,y) are both constant and equal to zero, and are therefore continuous. The functionf is not continuous at(0, 0), because, ifε < 1/2 andy =x² ≠ 0, we havef(x,y) = 1/2, even if|x| 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(0, 0) are also continuous. In fact, we have

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

forλ ≠ 0.

Thelimit at a point of a real-valued function of several real variables is defined as follows.^[1] Leta = (a₁,a₂, …,a_n) be a point intopological closure of the domainX of the functionf. The function,f has a limitL whenx tends towarda, denoted

${\displaystyle L=\underset{\mathit{x}\to \mathit{a}}{lim}f(\mathit{x}),}$

if the following condition is satisfied:For every positive real numberε > 0, there is a positive real numberδ > 0 such that

for allx in the domain such that

If the limit exists, it is unique. Ifa is in the interior of the domain, the limit exists if and only if the function is continuous ata. In this case, we have

${\displaystyle f(\mathit{a})=\underset{\mathit{x}\to \mathit{a}}{lim}f(\mathit{x}).}$

Whena is in theboundary of the domain off, and iff has a limit ata, the latter formula allows to "extend by continuity" the domain off toa.

Asymmetric function is a functionf that is unchanged when two variablesx_i andx_j are interchanged:

${\displaystyle f(\dots ,x_i,\dots ,x_j,\dots )=f(\dots ,x_j,\dots ,x_i,\dots )}$

wherei andj are each one of1, 2, …,n. For example:

${\displaystyle f(x,y,z,t)=t^2-x^2-y^2-z^2}$

is symmetric inx,y,z since interchanging any pair ofx,y,z leavesf unchanged, but is not symmetric in all ofx,y,z,t, since interchangingt withx ory orz gives a different function.

Suppose the functions

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

or more compactlyξ =ξ(x), are all defined on a domainX. As then-tuplex = (x₁,x₂, …,x_n) varies inX, a subset ofRⁿ, them-tupleξ = (ξ₁,ξ₂, …,ξ_m) varies in another regionΞ a subset ofR^m. To restate this:

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

Then, a functionζ of the functionsξ(x) defined onΞ,

is afunction composition defined onX,^[2] in other terms the mapping

Note the numbersm andn do not need to be equal.

For example, the function

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

defined everywhere onR² can be rewritten by introducing

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

which is also everywhere defined inR³ to obtain

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

Function composition can be used to simplify functions, which is useful for carrying outmultiple integrals and solvingpartial differential equations.

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

Partial derivatives can be defined with respect to each variable:

${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1}f(x_1,x_2,\dots ,x_n),\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_2}f(x_1,x_2,\dots x_n),\dots ,\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_n}f(x_1,x_2,\dots ,x_n).}$

Partial derivatives themselves are functions, each of which represents the rate of change off parallel to one of thex₁,x₂, …,x_n 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,y =f(x), itsordinary derivativedy/dx is geometrically the gradient of the tangent line to the curvey =f(x) 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:

${\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_1^2}f(x_1,x_2,\dots ,x_n),\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_1{x}_2}f(x_1,x_2,\dots x_n),\dots ,\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_n^2}f(x_1,x_2,\dots ,x_n).}$

Geometrically, they are related to the localcurvature 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 possiblestationary points: global or localmaxima, global or localminima, andsaddle points—the multidimensional analogue ofinflection points for real functions of one real variable. TheHessian matrix is a matrix of all the second order partial derivatives, which are used to investigate the stationary points of the function, important formathematical optimization.

In general, partial derivatives of higher orderp have the form:

${\displaystyle \frac{{\mathrm{\partial }}^p}{\mathrm{\partial }{x}_1^{p_1}\mathrm{\partial }{x}_2^{p_2}\cdots \mathrm{\partial }{x}_n^{p_n}}f(x_1,x_2,\dots ,x_n)\equiv \frac{{\mathrm{\partial }}^{p_1}}{\mathrm{\partial }{x}_1^{p_1}}\frac{{\mathrm{\partial }}^{p_2}}{\mathrm{\partial }{x}_2^{p_2}}\cdots \frac{{\mathrm{\partial }}^{p_n}}{\mathrm{\partial }{x}_n^{p_n}}f(x_1,x_2,\dots ,x_n)}$

wherep₁,p₂, …,p_n are each integers between0 andp such thatp₁ +p₂ + ⋯ +p_n =p, using the definitions of zeroth partial derivatives asidentity operators:

${\displaystyle \frac{{\mathrm{\partial }}^0}{\mathrm{\partial }{x}_1^0}f(x_1,x_2,\dots ,x_n)=f(x_1,x_2,\dots ,x_n),\dots ,\frac{{\mathrm{\partial }}^0}{\mathrm{\partial }{x}_n^0}f(x_1,x_2,\dots ,x_n)=f(x_1,x_2,\dots ,x_n).}$

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

A functionf(x) isdifferentiable in a neighborhood of a pointa if there is ann-tuple of numbers dependent ona in general,A(a) = (A₁(a),A₂(a), …,A_n(a)), so that:^[3]

${\displaystyle f(\mathit{x})=f(\mathit{a})+\mathit{A}(\mathit{a})\cdot (\mathit{x}-\mathit{a})+\alpha (\mathit{x})|\mathit{x}-\mathit{a}|}$

where${\displaystyle \alpha (\mathit{x})\to 0}$ as${\displaystyle |\mathit{x}-\mathit{a}|\to 0}$. This means that iff is differentiable at a pointa, thenf is continuous atx =a, although the converse is not true - continuity in the domain does not imply differentiability in the domain. Iff is differentiable ata then the first order partial derivatives exist ata and:

${\displaystyle {\frac{\mathrm{\partial }f(\mathit{x})}{\mathrm{\partial }{x}_i}|}_{\mathit{x}=\mathit{a}}=A_i(\mathit{a})}$

fori = 1, 2, …,n, which can be found from the definitions of the individual partial derivatives, so the partial derivatives off exist.

Assuming ann-dimensional analogue of a rectangularCartesian coordinate system, these partial derivatives can be used to form a vectoriallineardifferential operator, called thegradient (also known as "nabla" or "del") in this coordinate system:

${\displaystyle \mathrm{\nabla }f(\mathit{x})=\left(\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1},\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_2},\dots ,\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_n}\right)f(\mathit{x})}$

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

Then substituting the gradient∇f (evaluated atx =a) with a slight rearrangement gives:

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

where· denotes thedot product. This equation represents the best linear approximation of the functionf at all pointsx within a neighborhood ofa. Forinfinitesimal changes inf andx asx →a:

${\displaystyle df={\frac{\mathrm{\partial }f(\mathit{x})}{\mathrm{\partial }{x}_1}|}_{\mathit{x}=\mathit{a}}d{x}_1+{\frac{\mathrm{\partial }f(\mathit{x})}{\mathrm{\partial }{x}_2}|}_{\mathit{x}=\mathit{a}}d{x}_2+\cdots +{\frac{\mathrm{\partial }f(\mathit{x})}{\mathrm{\partial }{x}_n}|}_{\mathit{x}=\mathit{a}}d{x}_n=\mathrm{\nabla }f(\mathit{a})\cdot d\mathit{x}}$

which is defined as thetotaldifferential, or simplydifferential, off, ata. This expression corresponds to the total infinitesimal change off, by adding all the infinitesimal changes off in all thex_i directions. Also,df can be construed as acovector withbasis vectors as the infinitesimalsdx_i in each direction and partial derivatives off as the components.

Geometrically∇f is perpendicular to the level sets off, given byf(x) =c which for some constantc describes an(n − 1)-dimensional hypersurface. The differential of a constant is zero:

${\displaystyle df=(\mathrm{\nabla }f)\cdot d\mathit{x}=0}$

in whichdx is an infinitesimal change inx in the hypersurfacef(x) =c, and since the dot product of∇f anddx is zero, this means∇f is perpendicular todx.

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

If all first order partial derivatives evaluated at a pointa in the domain:

${\displaystyle {\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1}f(\mathit{x})|}_{\mathit{x}=\mathit{a}},{\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_2}f(\mathit{x})|}_{\mathit{x}=\mathit{a}},\dots ,{\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_n}f(\mathit{x})|}_{\mathit{x}=\mathit{a}}}$

exist and are continuous for alla in the domain,f has differentiability classC¹. In general, if all orderp partial derivatives evaluated at a pointa:

${\displaystyle {\frac{{\mathrm{\partial }}^p}{\mathrm{\partial }{x}_1^{p_1}\mathrm{\partial }{x}_2^{p_2}\cdots \mathrm{\partial }{x}_n^{p_n}}f(\mathit{x})|}_{\mathit{x}=\mathit{a}}}$

exist and are continuous, wherep₁,p₂, …,p_n, andp are as above, for alla in the domain, thenf is differentiable to orderp throughout the domain and has differentiability classC^p.

Iff is of differentiability classC^∞,f has continuous partial derivatives of all order and is calledsmooth. Iff is ananalytic function and equals itsTaylor series about any point in the domain, the notationC^ω denotes this differentiability class.

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

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

where each regionR₁,R₂, …,R_n is a subset of or all of the real line:

${\displaystyle R_1\subseteq \mathbb{R},R_2\subseteq \mathbb{R},\dots ,R_n\subseteq \mathbb{R},}$

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

${\displaystyle R=R_1\times R_2\times \cdots \times R_n,R\subseteq {\mathbb{R}}^n,}$

ann-dimensionalhypervolume. When evaluated, a definite integral is a real number if the integralconverges in the regionR 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 variabley =f(x) with respect tox has geometric interpretation as the area bounded by the curvey =f(x) and thex-axis. Multiple integrals extend the dimensionality of this concept: assuming ann-dimensional analogue of a rectangularCartesian coordinate system, the above definite integral has the geometric interpretation as then-dimensional hypervolume bounded byf(x) and thex₁,x₂, …,x_n 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: iff is somescalar density field andx are theposition vector coordinates, i.e. somescalar quantity per unitn-dimensional hypervolume, then integrating over the regionR gives the total amount of quantity inR. The more formal notions of hypervolume is the subject ofmeasure theory. Above we used theLebesgue measure, seeLebesgue integration for more on this topic.

With the definitions of multiple integration and partial derivatives, key theorems can be formulated, including thefundamental theorem of calculus in several real variables (namelyStokes' theorem),integration by parts in several real variables, thesymmetry of higher partial derivatives andTaylor's theorem for multivariable functions. Evaluating a mixture of integrals and partial derivatives can be done by using theoremdifferentiation under the integral sign.

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

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

into anm-tuple, or sometimes as acolumn vector orrow vector, respectively:

all treated on the same footing as anm-componentvector field, and use whichever form is convenient. All the above notations have a common compact notationy =f(x). The calculus of such vector fields isvector calculus. For more on the treatment of row vectors and column vectors of multivariable functions, seematrix calculus.

A real-valuedimplicit function of several real variables is not written in the form "y =f(…)". Instead, the mapping is from the spaceR^{n + 1} to thezero element inR (just the ordinary zero 0):

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

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

then we can always define:

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

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

For example, usinginterval notation, let

Choosing a 3-dimensional (3D) Cartesian coordinate system, this function describes the surface of a 3Dellipsoid centered at the origin(x,y,z) = (0, 0, 0) with constantsemi-major axesa,b,c, along the positivex,y andz axes respectively. In the casea =b =c =r, we have asphere of radiusr centered at the origin. Otherconic section examples which can be described similarly include thehyperboloid andparaboloid, more generally so can any 2D surface in 3D Euclidean space. The above example can be solved forx,y orz; however it is much tidier to write it in an implicit form.

For a more sophisticated example:

for non-zero real constantsA,B,C,ω, this function is well-defined for all(t,x,y,z), but it cannot be solved explicitly for these variables and written as "t =", "x =", etc.

Theimplicit function theorem of more than two real variables deals with the continuity and differentiability of the function, as follows.^[4] Letϕ(x₁,x₂, …,x_n) be a continuous function with continuous first order partial derivatives, and letϕ evaluated at a point(a,b) = (a₁,a₂, …,a_n,b) be zero:

${\displaystyle \varphi (\mathit{a},b)=0;}$

and let the first partial derivative ofϕ with respect toy evaluated at(a,b) be non-zero:

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

Then, there is an interval[y₁,y₂] containingb, and a regionR containing(a,b), such that for everyx inR there is exactly one value ofy in[y₁,y₂] satisfyingϕ(x,y) = 0, andy is a continuous function ofx so thatϕ(x,y(x)) = 0. Thetotal differentials of the functions are:

${\displaystyle dy=\frac{\mathrm{\partial }y}{\mathrm{\partial }{x}_1}d{x}_1+\frac{\mathrm{\partial }y}{\mathrm{\partial }{x}_2}d{x}_2+\cdots +\frac{\mathrm{\partial }y}{\mathrm{\partial }{x}_n}d{x}_n;}$

${\displaystyle d\varphi =\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{x}_1}d{x}_1+\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{x}_2}d{x}_2+\cdots +\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{x}_n}d{x}_n+\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }y}dy.}$

Substitutingdy into the latter differential andequating coefficients of the differentials gives the first order partial derivatives ofy with respect tox_i in terms of the derivatives of the original function, each as a solution of the linear equation

${\displaystyle \frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{x}_i}+\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }y}\frac{\mathrm{\partial }y}{\mathrm{\partial }{x}_i}=0}$

fori = 1, 2, …,n.

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 allowingcomplex values.

Iff(x₁, …,x_n) is such a complex valued function, it may be decomposed as

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

whereg andh 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:

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

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

Multivariable functions of real variables arise inevitably inengineering andphysics, becauseobservablephysical quantities are real numbers (with associatedunits anddimensions), and any one physical quantity will generally depend on a number of other quantities.

Examples incontinuum mechanics include the local massdensityρ of a mass distribution, ascalar field which depends on the spatial position coordinates (here Cartesian to exemplify),r = (x,y,z), and timet:

${\displaystyle \rho =\rho (\mathbf{r},t)=\rho (x,y,z,t)}$

Similarly for electriccharge density forelectrically charged objects, and numerous otherscalar potential fields.

Another example is thevelocity field, avector field, which has components of velocityv = (v_x,v_y,v_z) that are each multivariable functions of spatial coordinates and time similarly:

${\displaystyle \mathbf{v}(\mathbf{r},t)=\mathbf{v}(x,y,z,t)=[v_x(x,y,z,t),v_y(x,y,z,t),v_z(x,y,z,t)]}$

Similarly for other physical vector fields such aselectric fields andmagnetic fields, andvector potential fields.

Another important example is theequation of state inthermodynamics, an equation relatingpressureP,temperatureT, andvolumeV of a fluid, in general it has an implicit form:

${\displaystyle f(P,V,T)=0}$

The simplest example is theideal gas law:

${\displaystyle f(P,V,T)=PV-nRT=0}$

wheren is thenumber of moles, constant for a fixedamount of substance, andR thegas 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 ineconomics. 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 ofdemand 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. Inproducer 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 ofsupply functions for the various products; each of these functions has as its arguments the prices of the goods and of the factors of production.

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

In two-dimensionalfluid mechanics, specifically in the theory of thepotential flows used to describe fluid motion in 2d, thecomplex potential

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

is a complex valued function of the two spatial coordinatesx andy, and otherreal variables associated with the system. The real part is thevelocity potential and the imaginary part is thestream function.

Thespherical harmonics occur in physics and engineering as the solution toLaplace's equation, as well as theeigenfunctions of thez-componentangular momentum operator, which are complex-valued functions of real-valuedspherical polar angles:

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

Inquantum mechanics, thewavefunction is necessarily complex-valued, but is a function ofreal spatial coordinates (ormomentum components), as well as timet:

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

where each is related by aFourier transform.

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

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• Mathematical analysis
• Real numbers
• Multivariable calculus

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