Implicit function theorem

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 sub submatrices:

$X=\left(\partial {f_{i} \over \partial x_{j}}|_{(\mathbf {a} ,\mathbf {b} )}\right)_{ij}$

$Y=\left(\partial {f_{i} \over \partial y_{j}}|_{(\mathbf {a} ,\mathbf {b} )}\right)_{ij}$

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.