Function of several real variables

In mathematical analysis, a real multivariable function or real multivariate function is a function of several real numbers.

General definition

Functions in mathematics map elements (often numbers) from one set to another by a rule. The "from set" is called the domain and "to set" the codomain, and both of these must be specified with the rule to completely define a function. Set-builder notation can be used to precisely define which elements (often numbers) X and Y contain.

For n real independent variables x₁, x₂, x₃ ... x_n in some domain X, which can be all or part of the real n-dimensional space ℝⁿ, or set theoretically X is a subset of or equal to ℝⁿ:

$(x_{1},x_{2}\cdots x_{n})\in X\,,\quad X\subseteq \mathbb {R} ^{n}\,,$

and one dependent variable y in some codomain Y, which can be all or part of the real line ℝ, or set theoretically Y is a subset of or equal to ℝ:

$y\in Y\,,\quad Y\subseteq \mathbb {R}$

a multivariable function of these n real variables is defined as a mapping from X to Y, which has the notation:

$F:X\rightarrow Y$

where F alone denotes the rule, and:

$y=F(x_{1},x_{2}\cdots x_{n})$

is a number, calculated from x₁, x₂, x₃ ... x_n according to what F is. The domain and codomain must always be such that the function is well-defined. A common case is avoiding infinities caused by possible divisions by zero.

Tuple notation

Technically, since (x₁, x₂, x₃ ... x_n) is an n-tuple, we should write:

$y=F((x_{1},x_{2}\cdots x_{n}))$

but it is conventional, unambiguous, and notationally cleaner to simply write one set of brackets, not two. Another notational convention is to denote tuples in the same way as for vectors (vectors can be treated as tuples), to use boldface x = (x₁, x₂, x₃ ... x_n) or overline x = (x₁, x₂, x₃ ... x_n), then:

$y=F(\mathbf {x} )\,,\quad y=F({\overline {x}})$

Implicit functions

A real multivariable implicit function Φ cannot be written in the form "y = F(...)", but has the form:

$\Phi (x_{1},x_{2}\cdots x_{n},y)=0$

Examples

We could have:

$F=F(x,y)={\frac {1}{2xy}}{\sqrt {x^{2}+y^{2}}}$

$X=\{(x,y)\in \mathbb {R} ^{2}\,:\,x^{2}+y^{2}\leq 8\}$

$Y=\{x,y\in \mathbb {R} :x\neq 0,x\neq 0\}$

which takes in all points in a closed disk X of radius √8 in the plane ℝ², and returns a point in Y, a subset of or equal to ℝ. The function does not include the origin (x, y) = (0, 0), if it did then F would be ill-defined at that point.

As an example of substituting values:

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

however, we couldn't evaluate at, say

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

since the domain here does not contain these values of x and y.

Applications

Multivariable functions of real variables almost always arise inevitably in all forms of engineering and physics, because physical quantities are real numbers (with associated units and dimensions), and often depend on each other, and may vary in space and time.

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:

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

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:

$\mathbf {v} (\mathbf {r} ,t)=\mathbf {v} (x,y,z,t)=[v_{x}(x,y,z,t),v_{y}(x,y,z,t),v_{z}(x,y,z,t)]$

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

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:

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

(here F should not be confused for the Helmholtz free energy, F simply denotes a function of the variables P, V, T). The simplest example is the ideal gas law:

$F(P,V,T)=PV==0$

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.

References

• F. Ayres, E. Mendelson (2009). Calculus. Schuam's outline series (5th ed.). McGraw Hill. ISBN 978-0-07-150861-2.
• R. Wrede, M. R. Spiegel (2010). Advanced calculus. Schuam's outline series (3rd ed.). McGraw Hill. ISBN 978-0-07-162366-7.
• R. Penrose (2005). The Road to Reality. Vintage books. ISBN 978-00994-40680.