The Implicit Function Theorem

Theorem 1.2.1: The Implicit Function Theorem

The system of m smooth equations in m + n unknowns,

F[Y, Z] = C

for a constant vector C, defines  Y = g[Z]  as a smooth explicit function of  Z  near  (Y_0, Z_0)  provided  F[Y_0, Z_0] = C  and the m linear equations in m + n unknowns  (dY, dZ) given by the total differential,

DF[Y_0, Z_0] · (dY, dZ) = 0  ⇔  Underoverscript[Σ, j = 1, arg3] f_i^j[Y_0, Z_0] · dy_j + Underoverscript[Σ, k = 1, arg3] f_i^k[Y_0, Z_0] · dz_k = 0 ,   i = 1, ⋯, m

can be solved uniquely for  dY  as a linear function of  dZ (where  Y_0  and  Z_0  are fixed.)  In other words, suppose the  m × m matrix  B = (f_i^j[Y_0, Z_0]) _ (i = 1, ⋯, m)^(j = 1, ⋯, m) is invertible.  Then there is an open neighborhood    of  (Y_0, Z_0) (in m + n-space), an open neighborhood  V  of  Z_0  in  m-space, coordinate functions  g_i[Z], i = 1, ⋯, m, smooth on  , such that

{(Y, Z)∈U : F[Y, Z] = K} = {(Y, Z) : Z∈V & Y = G[Z]}

Created by Mathematica  (July 2, 2004)