EDB — 1Q0

[1Q0]Prerequisites:[1GD]↺↻.Let \(A⊂ ℝ^ n\) be open and \(f:A→ℝ\) in \(C^ 1\). Fix \(\overline x∈ A\) such that \(f(\overline x)=0\), and \(∇ f(\overline x)≠ 0\): by the implicit function theorem [1GD]↺↻ the set \(E=\{ f=0\} \) is a graph in a neighborhood of \(\overline x\), and the plane tangent to this graph is the set of \(x\) for which

Compare this result to Lemma 7.7.1 in the notes [ 3 ] : ”the gradient is orthogonal to the level sets” .