Implicit Function Theorem

In multivariable calculus, the implicit function theorem is a tool which allows relations to be converted to functions. It does this by representing the relation as the graph of a function. There may not be a single function whose graph is 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 (an implicit function) satisfies some mild conditions on its partial derivatives, then one can in principle solve this equation for, at least over some small interval. Geometrically, the locus defined by will overlap locally with the graph of a function (an explicit function, see article on implicit functions).

Read more about Implicit Function Theorem:  First Example, Statement of The Theorem, The Circle Example, Application: Change of Coordinates

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