Implicit function theorem

In <a href="/facts/Multivariable_calculus/PEs5SgUF">multivariable calculus</a>, the implicit function theorem is a tool that allows <a href="/facts/Relation_(mathematics)/HjblDaMy">relations</a> to be converted to <a href="/facts/Functions_of_several_real_variables/cyujR2q2">functions of several real variables</a>. It does so by representing the relation as the <a href="/facts/Graph_of_a_function/htYX0BGX">graph of a function</a>. 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 <a href="/facts/Domain_of_a_relation/r9BwGgaK">domain</a> of the relation. The implicit function theorem gives a sufficient condition to ensure that there is such a function.
More precisely, given a system of m equations fi (x1, ..., xn, y1, ..., ym) = 0, i = 1, ..., m (often abbreviated into F(x, y) = 0), the theorem states that, under a mild condition on the <a href="/facts/Partial_derivative/JWHsBkAu">partial derivatives</a> (with respect to each yi ) at a point, the m variables yi are differentiable functions of the xj in some <a href="/facts/Neighborhood_(mathematics)/XHKgrpkp">neighborhood</a> of the point. As these functions generally cannot be expressed in <a href="/facts/Closed_form_expression/y0xCXlSk">closed form</a>, they are implicitly defined by the equations, and this motivated the name of the theorem.
In other words, under a mild condition on the partial derivatives, the set of <a href="/facts/Zero_of_a_function/mlK3dhoT">zeros</a> of a system of equations is <a href="/facts/Local_property/sdBEwIy2">locally</a> the <a href="/facts/Graph_of_a_function/htYX0BGX">graph of a function</a>.

Implicit function theorem open-in-new

Implicit function theorem