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An implicit function is a function that is defined by an implicit equation, that relates one of the variables, considered as the value of the function, with the others considered as the arguments. [ 1 ] : 204–206 For example, the equation x 2 + y 2 − 1 = 0 {\displaystyle x^{2}+y^{2}-1=0} of the unit circle defines y as an implicit function ...
The unit circle can be specified as the level curve f(x, y) = 1 of the function f(x, y) = x 2 + y 2.Around point A, y can be expressed as a function y(x).In this example this function can be written explicitly as () =; in many cases no such explicit expression exists, but one can still refer to the implicit function y(x).
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However, the implicit function theorem gives conditions under which an implicit curve locally is given by the graph of a function (so in particular it has no self-intersections). If the defining relations are sufficiently smooth then, in such regions, implicit curves have well defined slopes, tangent lines, normal vectors, and curvature.
For such problems, to achieve given accuracy, it takes much less computational time to use an implicit method with larger time steps, even taking into account that one needs to solve an equation of the form (1) at each time step. That said, whether one should use an explicit or implicit method depends upon the problem to be solved.
Many other real functions are defined either by the implicit function theorem (the inverse function is a particular instance) or as solutions of differential equations. For example, the sine and the cosine functions are the solutions of the linear differential equation ″ + = such that
Formally, a function f:X→Y is said to be an implicit function (defined by the equation R(x,y)=0) if it satisfies the equation R(x,f(x)) = 0 for all x∈X. That brings R into the definition. An analogy: it doesn't make much sense to say "p is a characteristic polynomial", but it does make sense to say "p is the characteristic polynomial of the ...
A counterexample to this is provided by explicit function f(x,y)=x^2. The implicit function would be x(y)=0, but the explicit function doesn't satisfy the locally one-to-one property. I wrote this talk post in hope that in the future someone doesn't re-edit the mistake into the article. 193.40.13.174 10:27, 19 October 2022 (UTC)