Function of several real variables

The image of a function f(x₁, x₂, ..., x_n) is the set of all values of f when the n-tuple (x₁, x₂, ..., x_n) runs in the whole domain of f. For a continuous (see below for a definition) real-valued function which has a connected domain, the image is either an interval or a single value. In the latter case, the function is a constant function.

The preimage of a given real number c is called a level set. It is the set of the solutions of the equation f(x₁, x₂, ..., x_n) = c.

The domain of a function of several real variables is a subset of ℝⁿ that is sometimes, but not always, explicitly defined. In fact, if one restricts the domain X of a function f to a subset Y ⊂ X, one gets formally a different function, the restriction of f to Y, which is denoted f_|Y. In practice, it is often (but not always) not harmful to identify f and f_|Y, and to omit the subscript _|Y.

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

Moreover, many functions are defined in such a way that it is difficult to specify explicitly their domain. For example, given a function f, it may be difficult to specify the domain of the function ${\displaystyle g({\boldsymbol {x}})=1/f({\boldsymbol {x}}).}$ If f is a multivariate polynomial, (which has ${\displaystyle \mathbb {R} ^{n}}$ as a domain), it is even difficult to test whether the domain of g 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 (see Positive 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 number r, the constant function

${\displaystyle (x_{1},\ldots ,x_{n})\mapsto r}$

is everywhere defined.

• For every real number r and every function f, the function:

${\displaystyle rf:(x_{1},\ldots ,x_{n})\mapsto rf(x_{1},\ldots ,x_{n})}$

has the same domain as f (or is everywhere defined if r = 0).

• If f and g are two functions of respective domains X and Y such that X ∩ Y contains an open subset of ℝⁿ, then

${\displaystyle f\,g:(x_{1},\ldots ,x_{n})\mapsto f(x_{1},\ldots ,x_{n})\,g(x_{1},\ldots ,x_{n})}$

and

${\displaystyle f\,g:(x_{1},\ldots ,x_{n})\mapsto f(x_{1},\ldots ,x_{n})\,g(x_{1},\ldots ,x_{n})}$

are functions that have a domain containing X ∩ Y.

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

One may similarly define

${\displaystyle 1/f:(x_{1},\ldots ,x_{n})\mapsto 1/f(x_{1},\ldots ,x_{n}),}$

which is a function only if the set of the points (x₁, ...,x_n) in the domain of f such that f(x₁, ...,x_n) ≠ 0 contains an open subset of ℝⁿ. This constraint implies that the above two algebras are not fields.

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 the interior of the domain of the function f, we can fix the values of x₂, ..., x_n to a₂, ..., a_n respectively, to get a univariable function

${\displaystyle x\mapsto f(x,a_{2},\ldots ,a_{n}),}$

whose domain contains an interval centered at a₁. This function may also be viewed as the restriction of the function f to the line defined by the equations x_i = a_i, for i = 2, ...,n.

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

${\displaystyle x\mapsto f(a_{1}+c_{1}x,a_{2}+c_{2}x,\ldots ,a_{n}+c_{n}x),}$

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

For defining the continuity, it is useful to consider the distance function of ℝⁿ, which is an everywhere defined function of 2n real variables:

${\displaystyle d({\boldsymbol {x}},{\boldsymbol {y}})=d(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n})={\sqrt {(x_{1}-y_{1})^{2}+\cdots +(x_{n}-y_{n})^{2}}}}$

A function f is continuous at a point a = (a₁, ..., a_n) which is interior to its domain, if, for every positive real number ε, there is a positive real number φ such that |f(x) − f(a)| < ε for all x such that d(x a) < φ. In other words, φ may be chosen small enough for having the image by f of the ball of radius φ centered at a contained in the interval of length 2ε centered at f(a). A function is continuous if it is continuous at every point of its domain.

If a function is continuous at f(a), then all the univariate functions, that are obtained by fixing all the variables x_i but one at the value a_i, are continuous at f(a). The converse is false; this means that all these univariate functions may be continuous for a function that is not continuous at f(a). For an example, consider the function f such that f(0, 0) = 0, and is otherwise defined by

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

The functions x ↦ f(x, 0) and y ↦ f(0, y) are both constant and equal to zero, and are therefore continuous. The function f is not continuous at (0, 0), because, if ε < 1/2 and y = x² ≠ 0, we have f(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

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

for λ ≠ 0.

The limit at a point of a real-valued function of several real variables is defined as follows.[1] Let a = (a₁, a₂, ..., a_n) be a point in topological closure of the domain X of the function f. The function, f has a limit L when x tends toward a, denoted

${\displaystyle L=\lim _{{\boldsymbol {x}}\rightarrow {\boldsymbol {a}}}f({\boldsymbol {x}}),}$

if the following condition is satisfied: For every positive real number ε > 0, there is a positive real number δ > 0 such that

${\displaystyle |f({\boldsymbol {x}})-L|<\varepsilon }$

for all x in the domain such that

${\displaystyle d({\boldsymbol {x}},{\boldsymbol {a}})<\delta .}$

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

${\displaystyle f({\boldsymbol {a}})=\lim _{{\boldsymbol {x}}\rightarrow {\boldsymbol {a}}}f({\boldsymbol {x}}).}$

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

Partial derivatives can be defined with respect to each variable:

${\displaystyle {\frac {\partial }{\partial x_{1}}}f(x_{1},x_{2},\ldots ,x_{n})\,,\quad {\frac {\partial }{\partial x_{2}}}f(x_{1},x_{2},\ldots x_{n})\,,\ldots ,{\frac {\partial }{\partial x_{n}}}f(x_{1},x_{2},\ldots ,x_{n}).}$

Partial derivatives themselves are functions, each of which represents the rate of change of f parallel to one of the x₁, 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), its ordinary derivative dy/dx is geometrically the gradient of the tangent line to the curve y = 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:

${\displaystyle {\frac {\partial ^{2}}{\partial x_{1}^{2}}}f(x_{1},x_{2},\ldots ,x_{n})\,,\quad {\frac {\partial ^{2}}{\partial x_{1}x_{2}}}f(x_{1},x_{2},\ldots x_{n})\,,\ldots ,{\frac {\partial ^{2}}{\partial x_{n}^{2}}}f(x_{1},x_{2},\ldots ,x_{n}).}$

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

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

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

${\displaystyle {\frac {\partial ^{p}}{\partial x_{1}^{p_{1}}\partial x_{2}^{p_{2}}\ldots \partial x_{n}^{p_{n}}}}f(x_{1},x_{2},\ldots ,x_{n})\equiv {\frac {\partial ^{p_{1}}}{\partial x_{1}^{p_{1}}}}{\frac {\partial ^{p_{2}}}{\partial x_{2}^{p_{2}}}}\cdots {\frac {\partial ^{p_{n}}}{\partial x_{n}^{p_{n}}}}f(x_{1},x_{2},\ldots ,x_{n})}$

where p₁, p₂, ..., p_n are each integers between 0 and p such that p₁ + p₂ + ... + p_n = p, using the definitions of zeroth partial derivatives as identity operators:

${\displaystyle {\frac {\partial ^{0}}{\partial x_{1}^{0}}}f(x_{1},x_{2},\ldots ,x_{n})=f(x_{1},x_{2},\ldots ,x_{n})\,,\quad \ldots {\frac {\partial ^{0}}{\partial x_{n}^{0}}}f(x_{1},x_{2},\ldots ,x_{n})=f(x_{1},x_{2},\ldots ,x_{n})\,.}$

The number of possible partial derivatives increases with p, 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 p.

A function f(x) is differentiable in a neighborhood of a point a if there is an n-tuple of numbers dependent on a in general, A(a) = (A₁(a), A₂(a), ..., A_n(a)), so that:[3]

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

where α → 0 as |x − a| → 0. This means that if f is differentiable at a point a, then f is continuous at x = a, although the converse is not true - continuity in the domain does not imply differentiability in the domain. If f is differentiable at a then the first order partial derivatives exist at a and:

${\displaystyle \left.{\frac {\partial f({\boldsymbol {x}})}{\partial x_{i}}}\right|_{{\boldsymbol {x}}={\boldsymbol {a}}}=A_{i}({\boldsymbol {a}})}$

for i = 1, 2, ..., n, which can be found from the definitions of the individual partial derivatives, so the partial derivatives of f exist.

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

${\displaystyle \nabla f({\boldsymbol {x}})=\left({\frac {\partial }{\partial x_{1}}},{\frac {\partial }{\partial x_{2}}},\ldots ,{\frac {\partial }{\partial x_{n}}}\right)f({\boldsymbol {x}})}$

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

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

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

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

${\displaystyle df=\left.{\frac {\partial f({\boldsymbol {x}})}{\partial x_{1}}}\right|_{{\boldsymbol {x}}={\boldsymbol {a}}}dx_{1}+\left.{\frac {\partial f({\boldsymbol {x}})}{\partial x_{2}}}\right|_{{\boldsymbol {x}}={\boldsymbol {a}}}dx_{2}+\cdots \left.{\frac {\partial f({\boldsymbol {x}})}{\partial x_{n}}}\right|_{{\boldsymbol {x}}={\boldsymbol {a}}}dx_{n}=\nabla f({\boldsymbol {a}})\cdot d{\boldsymbol {x}}}$

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

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

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

in which dx is an infinitesimal change in x in the hypersurface f(x) = c, and since the dot product of ∇f and dx is zero, this means ∇f is perpendicular to dx.

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

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

${\displaystyle \left.{\frac {\partial }{\partial x_{1}}}f({\boldsymbol {x}})\right|_{{\boldsymbol {x}}={\boldsymbol {a}}}\,,\quad \left.{\frac {\partial }{\partial x_{2}}}f({\boldsymbol {x}})\right|_{{\boldsymbol {x}}={\boldsymbol {a}}}\,,\ldots ,\left.{\frac {\partial }{\partial x_{n}}}f({\boldsymbol {x}})\right|_{{\boldsymbol {x}}={\boldsymbol {a}}}}$

exist and are continuous for all a in the domain, f has differentiability class C¹. In general, if all order p partial derivatives evaluated at a point a:

${\displaystyle \left.{\frac {\partial ^{p}}{\partial x_{1}^{p_{1}}\partial x_{2}^{p_{2}}\ldots \partial x_{n}^{p_{n}}}}f({\boldsymbol {x}})\right|_{{\boldsymbol {x}}={\boldsymbol {a}}}}$

exist and are continuous, where p₁, p₂, ..., p_n, and p are as above, for all a in the domain, then f is differentiable to order p throughout the domain and has differentiability class C ^p.

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

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

${\displaystyle \int _{R_{n}}\cdots \int _{R_{2}}\int _{R_{1}}f(x_{1},x_{2},\ldots ,x_{n})\,dx_{1}dx_{2}\cdots dx_{n}\equiv \int _{R}f({\boldsymbol {x}})\,d^{n}{\boldsymbol {x}}}$

where each region R₁, R₂, ..., R_n is a subset of or all of the real line:

${\displaystyle R_{1}\subseteq \mathbb {R} \,,\quad R_{2}\subseteq \mathbb {R} \,,\ldots ,R_{n}\subseteq \mathbb {R} ,}$

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

${\displaystyle R=R_{1}\times R_{2}\times \ldots R_{n}\,,\quad R\subseteq \mathbb {R} ^{n}\,,}$

an n-dimensional hypervolume. When evaluated, a definite integral is a real number if the integral converges in the region R 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 y = f(x) with respect to x has geometric interpretation as the area bounded by the curve y = f(x) and the x-axis. Multiple integrals extend the dimensionality of this concept: assuming an n-dimensional analogue of a rectangular Cartesian coordinate system, the above definite integral has the geometric interpretation as the n-dimensional hypervolume bounded by f(x) and the x₁, 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: if f is some scalar density field and x are the position vector coordinates, i.e. some scalar quantity per unit n-dimensional hypervolume, then integrating over the region R gives the total amount of quantity in R. The more formal notions of hypervolume is the subject of measure theory. Above we used the Lebesgue measure, see Lebesgue integration for more on this topic.

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

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

${\displaystyle y_{1}=f_{1}(x_{1},x_{2},\ldots ,x_{n})\,,\quad y_{2}=f_{2}(x_{1},x_{2},\ldots ,x_{n})\,,\ldots ,y_{m}=f_{m}(x_{1},x_{2},\cdots x_{n})}$

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

${\displaystyle (y_{1},y_{2},\ldots ,y_{m})\leftrightarrow {\begin{bmatrix}f_{1}(x_{1},x_{2},\ldots ,x_{n})\\f_{2}(x_{1},x_{2},\cdots x_{n})\\\vdots \\f_{m}(x_{1},x_{2},\ldots ,x_{n})\end{bmatrix}}\leftrightarrow {\begin{bmatrix}f_{1}(x_{1},x_{2},\ldots ,x_{n})&f_{2}(x_{1},x_{2},\ldots ,x_{n})&\cdots &f_{m}(x_{1},x_{2},\ldots ,x_{n})\end{bmatrix}}}$

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

Examples in continuum mechanics include the local mass density ρ of a mass distribution, a scalar field which depends on the spatial position coordinates (here Cartesian to exemplify), r = (x, y, z), and time t:

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

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

Another example is the velocity field, a vector field, which has components of velocity v = (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 as electric fields and magnetic fields, and vector potential fields.

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

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

The simplest example is the ideal gas law:

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

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

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

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

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

${\displaystyle F(x,y,\ldots )=\varphi (x,y,\ldots )+i\psi (x,y,\ldots )}$

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

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

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

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

${\displaystyle \Psi =\Psi (\mathbf {r} ,t)=\Psi (x,y,z,t)\,,\quad \Phi =\Phi (\mathbf {p} ,t)=\Phi (p_{x},p_{y},p_{z},t)}$

where each is related by a Fourier transform.