Implicit function

English

Pronunciation

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Noun

1. Template:Lb A function defined by a (multivariable) implicit equation when one of the variables is regarded as the value of the function, especially where said equation is such that the value is not directly calculable from the other variables. Template:C

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Usage notes

• Definition notes:
  • An Template:M is a relation expressed in the form $R(x_1,\dots ,x_n)=0$, where $R$ is a function of several variables (often a polynomial). More broadly, the relation can be a set of simultaneous equations.
  • The expression not directly calculable here simply means that the implicit equation must be manipulated in order to solve for the chosen value. Such manipulation is not always possible, and, even if it is, the result may be an expression that is not computable in a finite number of "standard operations" (an ambiguous term). (See also Template:Pedia)
• Notes concerning related terms:
  • Given an arbitrary implicit equation $R(x_1,\dots ,x_{n-1},y)=0$, the selection of $y$ as the value (and thus the $x_i$ as arguments) does not by any means entail that the map $f:(x_1,\dots ,x_{n-1})\to y$ is single-valued (and thus a function). The nonstandard concept of multivalued function (or multifunction) is sometimes invoked. Additionally, the terms Template:M and Template:M may be used. In general, the solution space of the equation is some subset of ${ℝ}^{n-1}$. The Template:M deals, in part, with what constraints on the equation ensure that $f$ is a function.
    • For example, solving for $y$ in the implicit equation $x^2+y^2-1=0$ yields the formula $y=\pm \sqrt{1-x^2}$. This formula produces real number values only over the interval $x\in [-1,1]$, and across that interval has two branches corresponding to the cases $y\ge 0$ and $y\le 0$. Depending on the equation, the number of branches can be infinite.
  • The technique of Template:M enables the conversion of an implicit equation directly into a differential equation, which may reveal properties of an implicit function without its needing to be explicitly formulated.

Related terms

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Translations

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See also

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Further reading

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