domain

(noun)

the set of all possible mathematical entities (points) where a given function is defined

Related Terms

Examples of domain in the following topics:

  • Continuity

    • The function $f$ is continuous at some point $c$ of its domain if the limit of $f(x)$ as $x$ approaches $c$ through the domain of $f$ exists and is equal to $f(c)$.
    • The function $f$ is said to be continuous if it is continuous at every point of its domain.
    • If the point $c$ in the domain of $f$ is not a limit point of the domain, then this condition is vacuously true, since $x$ cannot approach $c$ through values not equal to $c$.

  • Double Integrals Over General Regions

    • Double integrals can be evaluated over the integral domain of any general shape.
    • But there is no reason to limit the domain to a rectangular area.
    • The integral domain can be of any general shape.
    • any line perpendicular to this axis that passes between these two values intersects the domain in an interval whose endpoints are given by the graphs of two functions, $\alpha$ and $\beta$.
    • This domain is normal with respect to both the $x$- and $y$-axes.

  • The Derivative as a Function

    • Let $f$ be a function that has a derivative at every point $a$ in the domain of $f$.
    • The derivative of $f$ collects all the derivatives of $f$ at all the points in the domain of $f$.
    • Sometimes $f$ has a derivative at most, but not all, points of its domain.
    • It is still a function, but its domain is strictly smaller than the domain of $f$.
    • Using this idea, differentiation becomes a function of functions: The derivative is an operator whose domain is the set of all functions that have derivatives at every point of their domain and whose range is a set of functions.

  • Double Integrals in Polar Coordinates

    • When domain has a cylindrical symmetry and the function has several specific characteristics, apply the transformation to polar coordinates.
    • In $R^2$, if the domain has a cylindrical symmetry and the function has several particular characteristics, you can apply the transformation to polar coordinates, which means that the generic points $P(x, y)$ in Cartesian coordinates switch to their respective points in polar coordinates.
    • This allows one to change the shape of the domain and simplify the operations.
    • Once the function is transformed and the domain evaluated, it is possible to define the formula for the change of variables in polar coordinates:

  • Improper Integrals

    • Integrals are also improper if the integrand is undefined at an interior point of the domain of integration, or at multiple such points.
    • It is often necessary to use improper integrals in order to compute a value for integrals which may not exist in the conventional sense (as a Riemann integral, for instance) because of a singularity in the function, or an infinite endpoint of the domain of integration.
    • The original definition of the Riemann integral does not apply to a function such as $\frac{1}{x^2}$ on the interval $[1, \infty]$, because in this case the domain of integration is unbounded.
    • The problem here is that the integrand is unbounded in the domain of integration (the definition requires that both the domain of integration and the integrand be bounded).
    • The integral may need to be defined on an unbounded domain.

  • Vector-Valued Functions

    • A vector function covers a set of multidimensional vectors at the intersection of the domains of $f$, $g$, and $h$.
    • For example, if you were to slice the three-dimensional shape perpendicular to the $z$-axis, the graph you would see would be of the function $z(t)=t$.The domain of a vector valued function is a domain that satisfies all of the component functions.
    • It can be found by taking the intersection of the individual component function domains.

  • Triple Integrals in Cylindrical Coordinates

    • In R3 the integration on domains with a circular base can be made by the passage in cylindrical coordinates; the transformation of the function is made by the following relation:
    • The domain transformation can be graphically attained, because only the shape of the base varies, while the height follows the shape of the starting region.
    • This method is convenient in case of cylindrical or conical domains or in regions where it is easy to individuate the $z$ interval and even transform the circular base and the function.
    • The function $f(x,y,z) = x^2 + y^2 + z$ is and as integration domain this cylinder:
    • Cylindrical coordinates are often used for integrations on domains with a circular base.

  • Triple Integrals in Spherical Coordinates

    • In $R^3$ some domains have a spherical symmetry, so it's possible to specify the coordinates of every point of the integration region by two angles and one distance.
    • The better integration domain for this passage is obviously the sphere.
    • Finally, you obtain the final integration formula: It's better to use this method in case of spherical domains and in case of functions that can be easily simplified, by the first fundamental relation of trigonometry, extended in $R^3$; in other cases it can be better to use cylindrical coordinates.
    • Integrate $f(x,y,z) = x^2 + y^2 + z^2$ over the domain $D = x^2 + y^2 + z^2 \le 16$.
    • Spherical coordinates are useful when domains in $R^3$ have spherical symmetry.

  • Inverse Functions

    • For this rule to be applicable, for a function whose domain is the set $X$ and whose range is the set $Y$, each element $y \in Y$ must correspond to no more than one $x \in X$; a function $f$ with this property is called one-to-one, or information-preserving, or an injection.
    • The function $f(x)=x^2$ may or may not be invertible, depending on the domain.
    • If the domain is the real numbers, then each element in the range $Y$ would correspond to two different elements in the domain $X$ ($\pm x$), and therefore $f$ would not be invertible.
    • If the domain consists of the non-negative numbers, then the function is injective and invertible.

  • Triple Integrals

    • If $T$ is a domain that is normal with respect to the xy-plane and determined by the functions $\alpha (x,y)$ and $\beta(x,y)$, then:
    • Integrate $f(x,y,z) = x^2 + y^2 + z^2$ over the domain $D = \left \{ x^2 + y^2 + z^2 \le 16 \right \}$.
    • Looking at the domain, it seems convenient to adopt the passage in spherical coordinates; in fact, the intervals of the variables that delimit the new $T$ region are obviously:
    • Example of domain in $R^3$ that is normal with respect to the $xy$-plane.