EDB — 1GQ

Exercises

1. [1GQ]Let \(A⊂ ℝ^ 3\) be an open set and suppose that \(f,g:A→ℝ\) is differentiable, and such that in \(p_ 0=(x_ 0,y_ 0,z_ 0)∈ A\) we have that \(∇ f(p_ 0),∇ g(p_ 0) \) are linearly independent and \(f(p_ 0)=g(p_ 0)=0\): show that the set \(E=\{ f=0,g=0\} \) is a curve in a neighborhood of \(p_ 0\).

   (Hint: consider that the vector product \(w=∇ f(p_ 0)× ∇ g(p_ 0)\) is nonzero if and only if the vectors are linearly independent — in fact it is formed by the determinants of the minors of the Jacobian matrix. Assuming without loss of generality that \(w_ 3≠ 0\), show that \(E\) is locally the graph of a function \((x,y)=𝛾(z)\).)

   Solution 1

   [1GR]↺↻