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mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
and its applications, a function of several real variables or real multivariate function is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
with more than one
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialecti ...
, with all arguments being
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
variables. This concept extends the idea of a
function of a real variable In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an interv ...
to several variables. The "input" variables take real values, while the "output", also called the "value of the function", may be real or
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
. 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 Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
of a function of variables is the
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of for which the function is defined. As usual, the domain of a function of several real variables is supposed to contain a nonempty
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * Open (Blues Image album), ''Open'' (Blues Image album), 1969 * Open (Gotthard album), ''Open'' (Gotthard album), 1999 * Open (C ...
subset of .


General definition

A real-valued function of real variables is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
that takes as input
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, commonly represented by the variables , for producing another real number, the ''value'' of the function, commonly denoted . For simplicity, in this article a real-valued function of several real variables will be simply called a function. To avoid any ambiguity, the other types of functions that may occur will be explicitly specified. Some functions are defined for all real values of the variables (one says that they are everywhere defined), but some other functions are defined only if the value of the variable are taken in a subset of , the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
of the function, which is always supposed to contain an
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * Open (Blues Image album), ''Open'' (Blues Image album), 1969 * Open (Gotthard album), ''Open'' (Gotthard album), 1999 * Open (C ...
subset of . In other words, a real-valued function of real variables is a function :f: X \to \R such that its domain is a subset of that contains a nonempty open set. An element of being an -
tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
(usually delimited by parentheses), the general notation for denoting functions would be . The common usage, much older than the general definition of functions between sets, is to not use double parentheses and to simply write . It is also common to abbreviate the -tuple by using a notation similar to that for vectors, like boldface , underline , or overarrow . This article will use bold. A simple example of a function in two variables could be: :\begin & V : X \to \R \\ & X = \left\ \\ & V(A,h) = \fracA h \end which is the
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
of a
cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...
with base area and height measured perpendicularly from the base. The domain restricts all variables to be positive since
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...
s and
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
s must be positive. For an example of a function in two variables: :\begin & z : \R^2 \to \R \\ & z(x,y) = ax + by \end where and are real non-zero constants. Using the
three-dimensional Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called '' parameters'') are required to determine the position of an element (i.e., point). This is the inform ...
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
, where the ''xy'' plane is the domain and the z axis is the codomain , one can visualize the image to be a two-dimensional plane, with a
slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
of in the positive x direction and a slope of in the positive y direction. The function is well-defined at all points in . The previous example can be extended easily to higher dimensions: :\begin & 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 for non-zero real constants , which describes a -dimensional
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
. The
Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
: :f(\boldsymbol)=\, \boldsymbol\, = \sqrt is also a function of ''n'' variables which is everywhere defined, while :g(\boldsymbol)=\frac is defined only for . For a non-linear example function in two variables: :\begin & z : X \to \R \\ & X = \left\ \\ & z(x,y) = \frac\sqrt \end which takes in all points in , a disk of radius "punctured" at the origin in the plane , and returns a point in . The function does not include the origin , if it did then would be ill-defined at that point. Using a 3d Cartesian coordinate system with the ''xy''-plane as the domain , and the z axis the codomain , the image can be visualized as a curved surface. The function can be evaluated at the point in : :z\left(2,\sqrt\right) = \frac\sqrt = \frac\sqrt \,, However, the function couldn't be evaluated at, say :(x,y) = (65,\sqrt) \, \Rightarrow \, x^2 + y^2 = (65)^2 + (\sqrt)^2 > 8 since these values of and do not satisfy the domain's rule.


Image

The
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
of a function is the set of all values of when the -tuple runs in the whole domain of . 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 In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image). Basic propertie ...
. The
preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
of a given real number is called a
level set In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is: : L_c(f) = \left\~, When the number of independent variables is two, a level set is calle ...
. It is the set of the solutions of the
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in F ...
.


Domain

The
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
of a function of several real variables is a subset of that is sometimes, but not always, explicitly defined. In fact, if one restricts the domain of a function to a subset , one gets formally a different function, the ''restriction'' of to , which is denoted f, _Y. In practice, it is often (but not always) not harmful to identify and f, _Y, and to omit the restrictor . Conversely, it is sometimes possible to enlarge naturally the domain of a given function, for example by continuity or by
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ...
. Moreover, many functions are defined in such a way that it is difficult to specify explicitly their domain. For example, given a function , it may be difficult to specify the domain of the function g(\boldsymbol) = 1/f(\boldsymbol). If is a
multivariate polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
, (which has \R^n as a domain), it is even difficult to test whether the domain of 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 In mathematics, a positive polynomial on a particular set is a polynomial whose values are positive on that set. Let ''p'' be a polynomial in ''n'' variables with real coefficients and let ''S'' be a subset of the ''n''-dimensional Euclidean ...
).


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 , the
constant function In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image). Basic propertie ...
(x_1,\ldots,x_n)\mapsto r is everywhere defined. * For every real number and every function , the function: rf:(x_1,\ldots,x_n)\mapsto rf(x_1,\ldots,x_n) has the same domain as (or is everywhere defined if ). * If and are two functions of respective domains and such that contains a nonempty open subset of , then f\,g:(x_1,\ldots,x_n)\mapsto f(x_1,\ldots,x_n)\,g(x_1,\ldots,x_n) and g\,f:(x_1,\ldots,x_n)\mapsto g(x_1,\ldots,x_n)\,f(x_1,\ldots,x_n) are functions that have a domain containing . It follows that the functions of variables that are everywhere defined and the functions of variables that are defined in some
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural a ...
of a given point both form commutative algebras over the reals (-algebras). This is a prototypical example of a
function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
. One may similarly define :1/f : (x_1,\ldots,x_n) \mapsto 1/f(x_1,\ldots,x_n), which is a function only if the set of the points in the domain of such that contains an open subset of . 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 is a point of the interior of the domain of the function , we can fix the values of to respectively, to get a univariable function :x \mapsto f(x, a_2, \ldots, a_n), whose domain contains an interval centered at . This function may also be viewed as the restriction of the function to the line defined by the equations for . Other univariable functions may be defined by restricting to any line passing through . These are the functions :x \mapsto f(a_1+c_1 x, a_2+c_2 x, \ldots, a_n+c_n x), where the 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 function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s 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 In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
and a
continuous map In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
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 , which is an everywhere defined function of real variables: :d(\boldsymbol,\boldsymbol)=d(x_1, \ldots, x_n, y_1, \ldots, y_n)=\sqrt A function is continuous at a point which is interior to its domain, if, for every positive real number , there is a positive real number such that for all such that . In other words, may be chosen small enough for having the image by of the ball of radius centered at contained in the interval of length centered at . A function is continuous if it is continuous at every point of its domain. If a function is continuous at , then all the univariate functions that are obtained by fixing all the variables except one at the value , are continuous at . The converse is false; this means that all these univariate functions may be continuous for a function that is not continuous at . For an example, consider the function such that , and is otherwise defined by :f(x,y) = \frac. The functions and are both constant and equal to zero, and are therefore continuous. The function is not continuous at , because, if and , we have , even if 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 are also continuous. In fact, we have : f(x, \lambda x) =\frac for . The
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
at a point of a real-valued function of several real variables is defined as follows. Let be a point in
topological closure In topology, the closure of a subset of points in a topological space consists of all points in together with all limit points of . The closure of may equivalently be defined as the union of and its boundary, and also as the intersection ...
of the domain of the function . The function, has a limit when tends toward , denoted :L = \lim_ f(\boldsymbol), if the following condition is satisfied: For every positive real number , there is a positive real number such that :, f(\boldsymbol) - L, < \varepsilon for all in the domain such that :d(\boldsymbol, \boldsymbol)< \delta. If the limit exists, it is unique. If is in the interior of the domain, the limit exists if and only if the function is continuous at . In this case, we have :f(\boldsymbol) = \lim_ f(\boldsymbol). When is in the
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
of the domain of , and if has a limit at , the latter formula allows to "extend by continuity" the domain of to .


Symmetry

A
symmetric function In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f ...
is a function that is unchanged when two variables and are interchanged: :f(\ldots, x_i,\ldots,x_j,\ldots) = f(\ldots, x_j,\ldots,x_i,\ldots) where and are each one of . For example: :f(x,y,z,t) = t^2 - x^2 - y^2 - z^2 is symmetric in since interchanging any pair of leaves unchanged, but is not symmetric in all of , since interchanging with or or gives a different function.


Function composition

Suppose the functions :\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), or more compactly , are all defined on a domain . As the -tuple varies in , a subset of , the -tuple varies in another region a subset of . To restate this: :\boldsymbol : X \to \Xi . Then, a function of the functions defined on , :\begin & \zeta : \Xi \to \R, \\ & \zeta = \zeta(\xi_1,\xi_2,\ldots,\xi_m), \end is a
function composition In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
defined on , in other terms the mapping :\begin & \zeta : X \to \R , \\ & \zeta = \zeta(\xi_1,\xi_2,\ldots,\xi_m) = f(x_1,x_2,\ldots,x_n). \end Note the numbers and do not need to be equal. For example, the function :f(x,y) = e^ sin 3(x-y) - \cos 2(x+y)/math> defined everywhere on 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 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 integral In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number ...
s and solving
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...
s.


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 Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one Variable (mathematics), variable to calculus with Function of several real variables, functions of several variables: the Differential calculus, di ...
.


Partial derivatives

Partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s can be defined with respect to each variable: :\frac f(x_1, x_2, \ldots, x_n)\,,\quad \frac f(x_1, x_2, \ldots x_n)\,,\ldots, \frac f(x_1, x_2, \ldots, x_n). Partial derivatives themselves are functions, each of which represents the rate of change of parallel to one of the 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, , its
ordinary derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
is geometrically the gradient of the tangent line to the curve 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 f(x_1, x_2, \ldots, x_n)\,,\quad \frac f(x_1, x_2, \ldots x_n)\,,\ldots, \frac f(x_1, x_2, \ldots, x_n) . Geometrically, they are related to the local
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
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 point In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" in ...
s: global or local maxima, global or local minima, and
saddle point In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the functi ...
s—the multidimensional analogue of
inflection point In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case ...
s for real functions of one real variable. The
Hessian matrix In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
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 Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
. In general, partial derivatives of higher order have the form: :\frac f(x_1, x_2, \ldots, x_n) \equiv \frac \frac \cdots \frac f(x_1, x_2, \ldots, x_n) where are each integers between and such that , using the definitions of zeroth partial derivatives as
identity operator Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film) ...
s: :\fracf(x_1, x_2, \ldots, x_n) = f(x_1, x_2, \ldots, x_n)\,,\quad \ldots,\, \fracf(x_1, x_2, \ldots, x_n)=f(x_1, x_2, \ldots, x_n)\,. The number of possible partial derivatives increases with , 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 .


Multivariable differentiability

A function is differentiable in a neighborhood of a point if there is an -tuple of numbers dependent on in general, , so that: :f(\boldsymbol) = f(\boldsymbol) + \boldsymbol(\boldsymbol)\cdot(\boldsymbol-\boldsymbol) + \alpha(\boldsymbol x), \boldsymbol-\boldsymbol, where as . This means that if is differentiable at a point , then is continuous at , although the converse is not true - continuity in the domain does not imply differentiability in the domain. If is differentiable at then the first order partial derivatives exist at and: :\left.\frac\_ = A_i (\boldsymbol) for , which can be found from the definitions of the individual partial derivatives, so the partial derivatives of exist. Assuming an -dimensional analogue of a rectangular
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
, these partial derivatives can be used to form a vectorial
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
, called the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
(also known as " nabla" or "
del Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes ...
") in this coordinate system: :\nabla f(\boldsymbol) = \left(\frac, \frac, \ldots, \frac \right) f(\boldsymbol) used extensively in
vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
, because it is useful for constructing other differential operators and compactly formulating theorems in vector calculus. Then substituting the gradient (evaluated at with a slight rearrangement gives: :f(\boldsymbol) - f(\boldsymbol)= \nabla f(\boldsymbol)\cdot(\boldsymbol-\boldsymbol) + \alpha , \boldsymbol-\boldsymbol, where denotes the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
. This equation represents the best linear approximation of the function at all points within a neighborhood of . For infinitesimal changes in and as : :df = \left.\frac\_dx_1 + \left.\frac\_dx_2 + \dots + \left.\frac\_dx_n = \nabla f(\boldsymbol) \cdot d\boldsymbol which is defined as the total differential, or simply differential, of , at . This expression corresponds to the total infinitesimal change of , by adding all the infinitesimal changes of in all the directions. Also, can be construed as a
covector In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the ...
with
basis vectors In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as componen ...
as the infinitesimals in each direction and partial derivatives of as the components. Geometrically is perpendicular to the level sets of , given by which for some constant describes an -dimensional hypersurface. The differential of a constant is zero: :df = (\nabla f) \cdot d \boldsymbol = 0 in which is an infinitesimal change in in the hypersurface , and since the dot product of and is zero, this means is perpendicular to . In arbitrary
curvilinear coordinate system In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inve ...
s in dimensions, the explicit expression for the gradient would not be so simple - there would be scale factors in terms of the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...
for that coordinate system. For the above case used throughout this article, the metric is just the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
and the scale factors are all 1.


Differentiability classes

If all first order partial derivatives evaluated at a point in the domain: :\left.\frac f(\boldsymbol)\_\,,\quad \left.\frac f(\boldsymbol)\_\,,\ldots, \left.\frac f(\boldsymbol)\_ exist and are continuous for all in the domain, has differentiability class . In general, if all order partial derivatives evaluated at a point : :\left.\frac f(\boldsymbol)\_ exist and are continuous, where , and are as above, for all in the domain, then is differentiable to order throughout the domain and has differentiability class . If is of differentiability class , has continuous partial derivatives of all order and is called ''
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
''. If is an ''
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
'' and equals its
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
about any point in the domain, the notation denotes this differentiability class.


Multiple integration

Definite integration can be extended to
multiple integration In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number ...
over the several real variables with the notation; :\int_ \cdots \int_ \int_ f(x_1, x_2, \ldots, x_n) \, dx_1 dx_2\cdots dx_n \equiv \int_R f(\boldsymbol) \, d^n\boldsymbol where each region is a subset of or all of the real line: :R_1 \subseteq \mathbb \,, \quad R_2 \subseteq \mathbb \,, \ldots , R_n \subseteq \mathbb, 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 \,,\quad R \subseteq \mathbb^n \,, an -dimensional hypervolume. When evaluated, a definite integral is a real number if the integral converges in the region 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 with respect to has geometric interpretation as the area bounded by the curve and the -axis. Multiple integrals extend the dimensionality of this concept: assuming an -dimensional analogue of a rectangular
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
, the above definite integral has the geometric interpretation as the -dimensional hypervolume bounded by and the 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 is some
scalar density In mathematics, a relative scalar (of weight ''w'') is a scalar-valued function whose transform under a coordinate transform, : \bar^j = \bar^j(x^i) on an ''n''-dimensional manifold obeys the following equation : \bar(\bar^j) = J^w f(x^i) ...
field and are the
position vector In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or ...
coordinates, i.e. some scalar quantity per unit ''n''-dimensional hypervolume, then integrating over the region gives the total amount of quantity in . The more formal notions of hypervolume is the subject of
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
. Above we used the
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wi ...
, see
Lebesgue integration In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Le ...
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 The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, ...
in several real variables (namely
Stokes' theorem Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
),
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivat ...
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 In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form \int_^ f(x,t)\,dt, where -\infty < a(x), b(x) < \infty and the integral are
.


Vector calculus

One can collect a number of functions each of several real variables, say :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) into an -tuple, or sometimes as a
column vector In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, c ...
or
row vector In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, c ...
, respectively: :(y_1, y_2, \ldots, y_m) \leftrightarrow \begin 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 \leftrightarrow \begin 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 all treated on the same footing as an -component vector field, and use whichever form is convenient. All the above notations have a common compact notation . The calculus of such vector fields is
vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
. For more on the treatment of row vectors and column vectors of multivariable functions, see
matrix calculus In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a ...
.


Implicit functions

A real-valued
implicit function In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit func ...
of several real variables is not written in the form "". Instead, the mapping is from the space to the
zero element In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An additive identi ...
in (just the ordinary zero 0): :\begin & \phi: \R^ \to \ \\ & \phi(x_1, x_2, \ldots, x_n, y) = 0 \end 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, \ldots, x_n) then we can always define: : \phi(x_1, x_2, \ldots, x_n, y) = y - f(x_1, x_2, \ldots, 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 In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Othe ...
, let :\begin & \phi : X \to \ \\ & \phi(x,y,z) = \left(\frac\right)^2 + \left(\frac\right)^2 + \left(\frac\right)^2 - 1 = 0 \\ & X = a,a\times b,b\times c,c= \left\ . \end Choosing a 3-dimensional (3D) Cartesian coordinate system, this function describes the surface of a 3D
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as th ...
centered at the origin with constant semi-major axes , along the positive ''x'', ''y'' and ''z'' axes respectively. In the case , we have a
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
of radius centered at the origin. Other
conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
examples which can be described similarly include the
hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...
and
paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plan ...
, more generally so can any 2D surface in 3D Euclidean space. The above example can be solved for , or ; however it is much tidier to write it in an implicit form. For a more sophisticated example: :\begin & \phi : \R^4 \to \ \\ & \phi(t,x,y,z) = C tz e^ + A \sin(3\omega t) \left(x^2z - B y^6\right) = 0 \end for non-zero real constants , this function is well-defined for all , but it cannot be solved explicitly for these variables and written as "", "", etc. The implicit function theorem of more than two real variables deals with the continuity and differentiability of the function, as follows. Let be a continuous function with continuous first order partial derivatives, and let ''ϕ'' evaluated at a point be zero: :\phi(\boldsymbol, b) = 0; and let the first partial derivative of with respect to evaluated at be non-zero: :\left.\frac\_ \neq 0 . Then, there is an interval containing , and a region containing , such that for every in there is exactly one value of in satisfying , and is a continuous function of so that . The
total differential In calculus, the differential represents the principal part of the change in a function ''y'' = ''f''(''x'') with respect to changes in the independent variable. The differential ''dy'' is defined by :dy = f'(x)\,dx, where f'(x) is the ...
s of the functions are: :dy=\fracdx_1 + \fracdx_2 + \dots + \fracdx_n ; :d\phi=\fracdx_1 + \fracdx_2 + \dots + \fracdx_n + \fracdy . Substituting into the latter differential and equating coefficients of the differentials gives the first order partial derivatives of with respect to in terms of the derivatives of the original function, each as a solution of the linear equation :\frac + \frac\frac = 0 for .


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 Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
values. If is such a complex valued function, it may be decomposed as :f(x_1,\ldots, x_n)=g(x_1,\ldots, x_n)+ih(x_1,\ldots, x_n), where and 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 \left ln\left(x+iy- ae^\right) - \ln\left(x+iy + ae^\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 Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
and
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, because
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phy ...
physical quantities A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For exam ...
are real numbers (with associated
units Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...
and
dimensions In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
), 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 Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such mo ...
include the local mass
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
of a mass distribution, a
scalar field In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical quantity ...
which depends on the spatial position coordinates (here Cartesian to exemplify), , and time : :\rho = \rho(\mathbf,t) = \rho(x,y,z,t) Similarly for electric
charge density In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system i ...
for
electrically charged Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectiv ...
objects, and numerous other
scalar potential In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in trav ...
fields. Another example is the
velocity field In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
, a vector field, which has components of velocity that are each multivariable functions of spatial coordinates and time similarly: :\mathbf (\mathbf,t) = \mathbf(x,y,z,t) =
_x(x,y,z,t), v_y(x,y,z,t), v_z(x,y,z,t) X, or x, is the twenty-fourth and third-to-last Letter (alphabet), letter in the Latin alphabet, used in the English alphabet, modern English alphabet, the alphabets of other western European languages and others worldwide. Its English a ...
/math> Similarly for other physical vector fields such as
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
s and
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
s, and
vector potential In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field. Formally, given a vector field v, a ''vecto ...
fields. Another important example is the
equation of state In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or intern ...
in
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...
, an equation relating
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
,
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
, and
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
of a fluid, in general it has an implicit form: :f(P, V, T) = 0 The simplest example is the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first s ...
: :f(P, V, T) = PV - nRT = 0 where is the number of moles, constant for a fixed
amount of substance In chemistry, the amount of substance ''n'' in a given sample of matter is defined as the quantity or number of discrete atomic-scale particles in it divided by the Avogadro constant ''N''A. The particles or entities may be molecules, atoms, io ...
, and the
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
. 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 Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
. In the underpinnings of consumer theory,
utility As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosophe ...
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 function In economics, a demand curve is a graph depicting the relationship between the price of a certain commodity (the ''y''-axis) and the quantity of that commodity that is demanded at that price (the ''x''-axis). Demand curves can be used either for ...
s, 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 function In economics, supply is the amount of a resource that firms, producers, labourers, providers of financial assets, or other economic agents are willing and able to provide to the marketplace or to an individual. Supply can be in produced goods, l ...
s 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 In electrical engineering, impedance is the opposition to alternating current presented by the combined effect of resistance and reactance in a circuit. Quantitatively, the impedance of a two-terminal circuit element is the ratio of the compl ...
, complex permittivity, complex permeability, and
complex refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
. These are also functions of real variables, such as frequency or time, as well as temperature. In two-dimensional
fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
, specifically in the theory of the potential flows used to describe fluid motion in 2d, the complex potential :F(x,y,\ldots) = \varphi(x,y,\ldots) + i\psi(x,y,\ldots) is a complex valued function of the two spatial coordinates and , and other ''real'' variables associated with the system. The real part is the velocity potential and the imaginary part is the
stream function The stream function is defined for incompressible ( divergence-free) flows in two dimensions – as well as in three dimensions with axisymmetry. The flow velocity components can be expressed as the derivatives of the scalar stream function. T ...
. The
spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form ...
occur in physics and engineering as the solution to
Laplace's equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \na ...
, as well as the
eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s of the ''z''-component
angular momentum operator In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum p ...
, which are complex-valued functions of real-valued spherical polar angles: :Y^m_\ell = Y^m_\ell(\theta,\phi) In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
is necessarily complex-valued, but is a function of ''real'' spatial coordinates (or
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
components), as well as time : :\Psi = \Psi(\mathbf,t) = \Psi(x,y,z,t)\,,\quad \Phi = \Phi(\mathbf,t) = \Phi(p_x,p_y,p_z,t) where each is related by a
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
.


See also

*
Real coordinate space In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vector ...
*
Real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
*
Complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
*
Function of several complex variables The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variab ...
*
Scalar field In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical quantity ...
s


References

* * * * * * * * {{Authority control Mathematical analysis Real numbers Multivariable calculus