21 Surfaces[1PZ]

E417

[1Q0]Prerequisites:398.Let $A \subset ℝ^{n}$ be open and $f : A \to ℝ$ in $C^{1}$ . Fix $\overset{―}{x} \in A$ such that $f (\overset{―}{x}) = 0$ , and $\nabla f (\overset{―}{x}) \neq 0$ : by the implicit function theorem 398 the set $E = {f = 0}$ is a graph in a neighborhood of $\overset{―}{x}$ , and the plane tangent to this graph is the set of $x$ for which

⟨ x - \overset{―}{x}, \nabla f (\overset{―}{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’]

E417

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

Hidden solution: [UNACCESSIBLE UUID ’1Q3’]

E417

[1Q4]Let $a > 0$ . Show that the equation $\sqrt{x} + \sqrt{y} + \sqrt{z} = \sqrt{a}$ defines a regular surface inside the first octant ${x > 0, y > 0, z > 0}$ . 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’]

E417

[1Q8]Fix $a > 0, b > 0, c > 0$ . Determine a plane tangent to the ellipsoid

x^{2} / a^{2} + y^{2} / b^{2} + z^{2} / c^{2} = 1

at a point with $x, y, z > 0$ , so that the tetrahedron bounded by this plane and the coordinated planes has minimum volume.

Hidden solution: [UNACCESSIBLE UUID ’1Q9’]