Implicit function theorem - Definition and Overview

In mathematics, in the field of calculus of several variables, the implicit function theorem says that for a suitable set of equations, some of the variables are defined as a function of the others.

More precisely, that if f:R^m+n→Rⁿ, a is in R^m and b is in Rⁿ, with f(a,b)=0; if we take the Jacobian of f, and split it into submatrices:

<math>X=\left({\partial f_i \over \partial x_j}|_{(\mathbf{a},\mathbf{b})}\right)_{ij}<math>

<math>Y=\left({\partial f_i \over \partial y_j}|_{(\mathbf{a},\mathbf{b})}\right)_{ij}<math>

If Y is invertible (ie., the determinant Y is nonzero), f(x, y)=0 defines y as a function of x near (a,b), or that there exists a function such that g(b)=a, with its Jacobian at a being -Y^-1X.