22 Surfaces[1PZ]

E418

[1Q0]Prerequisites:399.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 399 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

\[ ⟨ x-\overline x,∇ f(\overline x) ⟩=0~ ~ . \]

Compare this result to Lemma 7.7.1 in the notes [ 2 ] : ”the gradient is orthogonal to the level sets” . Hidden solution: [UNACCESSIBLE UUID ’1Q1’]

E418

[1Q2]Given \(m{\gt}0\), show that the relation \(xyz=m^ 3\) defines a surface in \(ℝ^ 3\). Prove that the planes tangent to the surface at the points of the first octant \(\{ x{\gt}0,y{\gt}0,z{\gt}0\} \) form with the coordinate planes of \(ℝ^ 3\) a tetrahedron of constant volume.

Hidden solution: [UNACCESSIBLE UUID ’1Q3’]

E418

[1Q4]Let \(a{\gt}0\). Show that the equation \(\sqrt x + \sqrt y + \sqrt z = \sqrt a\) defines a regular surface inside the first octant \(\{ x{\gt}0,y{\gt}0,z{\gt}0\} \). Prove that planes tangent to the surface cut the three coordinate axes at three points, the sum of whose distances from the origin is constant.

Hidden solution: [UNACCESSIBLE UUID ’1Q5’]

E418

[1Q8]Fix \(a{\gt}0,b{\gt}0,c{\gt}0\). Determine a plane tangent to the ellipsoid

\[ x^ 2/a^ 2 + y^ 2/b^ 2 + z^ 2 / c^ 2 = 1 \]

at a point with \(x,y,z{\gt}0\), so that the tetrahedron bounded by this plane and the coordinated planes has minimum volume.

Hidden solution: [UNACCESSIBLE UUID ’1Q9’]