Implicit Function Theorem:

Let \(U \subseteq \mathbb{R}^n \times \mathbb{R}^k\) be an open subset. \((x,y)= (x_1,... ,x_n ,y_1 ,... , y_k)\) be elements of that set. Suppose \(\Phi : U \rightarrow \mathbb{R}^k\) is smooth, \((a,b) \in U, c = \Phi(a,b)\).

If \((\frac{\partial \Phi_i}{\partial y_j}(a,b))_{k \times k}\) is nonsingular, then there exists neighborhoods, \(V_0 \subseteq \mathbb{R}^n\) of \(a\) and \(W_0 \subseteq \mathbb{R}^k\) of \(b\) and a smooth \(F:V_0 \rightarrow W_0\) such that \(\Phi ^{-1}(c) \cap (V_0 \times W_0)\) is the graph of \(F\).

Meaning: \(\Phi(x,y) = c \) for \((x,y) \in (V_0,W_0) \iff y = F(x)\)