Implicit function theorem

In multivariable calculus, the implicit function theorem is a theorem that provides sufficient conditions under which a planar curve specified by ${\displaystyle F(x,y)=0}$ can also be specified as the graph of a function ${\displaystyle f}$, so that for each point ${\displaystyle (x,y)}$ on part of the curve, one has ${\displaystyle y=f(x)}$. An example is the unit circle, whose points ${\displaystyle (x,y)}$ satisfy ${\displaystyle x^{2}+y^{2}-1=0}$, which can locally be solved (if ${\displaystyle y>0}$) by ${\displaystyle y={\sqrt {1-x^{2}}}}$, expressing the top semicircle as a graph. It is not always possible to solve the equation ${\displaystyle F(x,y)=0}$ for ${\displaystyle y}$ algebraically, and the implicit function theorem gives analytic conditions under which there exists a function ${\displaystyle f}$ whose graph belongs to the given curve, and, in some formulations, also gives a way of constructing approximations to ${\displaystyle f}$.

More generally, given a system of m equations f_i (x₁, ..., x_n, y₁, ..., y_m) = 0, i = 1, ..., m (often abbreviated into F(x, y) = 0), the theorem states that, under a mild condition on the partial derivatives (with respect to each y_i ) at a point, the m variables y_i are differentiable functions of the x_j in some neighbourhood of the point. As these functions generally cannot be expressed in closed form, 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 zeros of a system of equations is locally the graph of a function.