As usual, the domain of a function of several real variables is supposed to contain a nonempty open subset of
The common usage, much older than the general definition of functions between sets, is to not use double parentheses and to simply write f(x1, x2, …, xn).
It is also common to abbreviate the n-tuple (x1, x2, …, xn) by using a notation similar to that for vectors, like boldface x, underline x, or overarrow x→.
Using the three-dimensional Cartesian coordinate system, where the xy plane is the domain R2 and the z axis is the codomain R, one can visualize the image to be a two-dimensional plane, with a slope of a in the positive x direction and a slope of b in the positive y direction.
The previous example can be extended easily to higher dimensions: for p non-zero real constants a1, a2, …, ap, which describes a p-dimensional hyperplane.
Using a 3d Cartesian coordinate system with the xy-plane as the domain R2, and the z axis the codomain R, the image can be visualized as a curved surface.
The image of a function f(x1, x2, …, xn) is the set of all values of f when the n-tuple (x1, x2, …, xn) 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.
It is the set of the solutions of the equation f(x1, x2, …, xn) = c. The domain of a function of several real variables is a subset of Rn that is sometimes, but not always, explicitly defined.
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.
For defining the continuity, it is useful to consider the distance function of Rn, which is an everywhere defined function of 2n real variables: A function f is continuous at a point a = (a1, …, an) 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.
[1] Let a = (a1, a2, …, an) 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 if the following condition is satisfied: For every positive real number ε > 0, there is a positive real number δ > 0 such that for all x in the domain such that If the limit exists, it is unique.
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: Geometrically, they are related to the local curvature of the function's image at all points in the domain.
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.
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: 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: where · denotes the dot product.
Also, df can be construed as a covector with basis vectors as the infinitesimals dxi in each direction and partial derivatives of f as the components.
If f is an analytic function and equals its Taylor series about any point in the domain, the notation Cω denotes this differentiability class.
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.
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.
For example, using interval notation, let Choosing a 3-dimensional (3D) Cartesian coordinate system, this function describes the surface of a 3D ellipsoid centered at the origin (x, y, z) = (0, 0, 0) with constant semi-major axes a, b, c, along the positive x, y and z axes respectively.
Other conic section examples which can be described similarly include the hyperboloid and paraboloid, more generally so can any 2D surface in 3D Euclidean space.
For a more sophisticated example: for non-zero real constants A, 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.
The total differentials of the functions are: Substituting dy into the latter differential and equating coefficients of the differentials gives the first order partial derivatives of y with respect to xi in terms of the derivatives of the original function, each as a solution of the linear equation for i = 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 allowing complex values.
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: Similarly for electric charge density for electrically charged objects, and numerous other scalar 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: The simplest example is the ideal gas law: where n is the number of moles, constant for a fixed amount of substance, and R the gas constant.
In two-dimensional fluid mechanics, specifically in the theory of the potential flows used to describe fluid motion in 2d, the complex potential is a complex valued function of the two spatial coordinates x and y, and other real variables associated with the system.
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: 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: where each is related by a Fourier transform.