Implicit function theorem

In mathematics, more specifically in multivariable calculus, the implicit function theorem is a tool that allows relations to be converted to functions of several real variables. It does so by representing the relation as the graph of a function. There may not be a single function whose graph can represent the entire relation, but there may be such a function on a restriction of the domain of the relation. The implicit function theorem gives a sufficient condition to ensure that there is such a function.

The theorem states that if the equation F(x₁, ..., x_n, y₁, ..., y_m) = F(x, y) = 0 satisfies some mild conditions on its partial derivatives, then one can in principle (though not necessarily with an analytic expression) express the m variables y_i in terms of the n variables x_j as y_i = f_i(x), at least in some disk. Then each of these implicit functions f_i(x), implied by F(x, y) = 0, is such that geometrically the locus defined by F(x, y) = 0 will coincide locally (that is in that disk) with the hypersurface given by y = f(x).

Augustin-Louis Cauchy (1789–1857) is credited with the first rigorous form of the implicit function theorem. Ulisse Dini (1845–1918) generalized the real-variable version of the implicit function theorem to the context of functions of any number of real variables.

If we define the function ${\displaystyle f(x,y)=x^{2}+y^{2}}$, then the equation f(x, y) = 1 cuts out the unit circle as the level set {(x, y)| f(x, y) = 1}. There is no way to represent the unit circle as the graph of a function of one variable y = g(x) because for each choice of x ∈ (−1, 1), there are two choices of y, namely ${\displaystyle \pm {\sqrt {1-x^{2}}}}$.

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