Homework 9

Due Date: Friday, November 7

1. Let \(S\) be the surface parameterized by \(\vec{X}(u,v) = (u^2+6v,3u+9v, u^2 + 2u +v^2)\).
   1. Find the critical point for the function \(\vec{X}\).
   2. Find a point on \(S\) whose tangent plane is parallel to the \(xy\)-plane.
2. Let \(P\) be the parabola \(x = z^2 - 1\) in the \(xz\)-plane, and let \(S\) be the surface obtained by rotating \(P\) around the \(z\)-axis.
   1. Find a parametrization \(\vec{X} \colon \mathbb{R}^2\to\mathbb{R}^3\) for the surface \(S\).
   2. Find the critical points for the function \(\vec{X}\). What is happening at the corresponding points on the surface?
3. Let \(\vec{F}\colon\mathbb{R}^3\to\mathbb{R}^2\) be the function \(\vec{F}(x,y,z) = \bigl(x,x^2+y^2+z^2\bigr)\).
   1. Find the set of critical points for \(\vec{F}\).
   2. Let \(F_1(x,y,z) = x\) and \(F_2(x,y,z) = x^2+y^2+z^2\) be the component functions of \(\vec{F}\). What are the level surfaces for these functions?
   3. What is the geometric relationship between the level surfaces for \(F_1\) and \(F_2\) at the critical points? Explain this phenomenon.
4. Let \(\vec{F}\colon\mathbb{R}^2\to\mathbb{R}^2\) be the function \(\vec{F}(x,y) = \bigl(2x^3 + y^3,x^2 + 2y^2\bigr)\). Then the set of critical points for \(\vec{F}\) is the union of three lines. Find the equations of these lines.
5. Let \(\vec{X}\colon\mathbb{R}^2 \to \mathbb{R}^3\) be a regular parametrization of a surface \(S\), and suppose that \(\displaystyle\vec{X}(3,9) = (5,3,1)\) and \(\displaystyle d\vec{X}(3,9) \;=\; \begin{bmatrix}1 & 1 \\ 2 & 1 \\ 2 & 0\end{bmatrix}\).
   1. Find the angle between the \(u\) and \(v\) coordinate lines for \(S\) at the point \((5,3,1)\).
   2. Find the equation of the tangent plane to \(S\) at the point \((5,3,1)\).
   3. Find \(\vec{x}\,'(9)\) if \(\vec{x}\) is the curve \(\vec{x}(t) = \vec{X}(3,t)\).
   4. Find \(\vec{y}\,'(3)\) if \(\vec{y}\) is the curve \(\vec{y}(t) = \vec{X}(t,t^2)\).
   5. Estimate the value of \(\vec{X}(3.02,9.05)\).
6. Let \(S\) be the helicoid \(\vec{X}(u,v) = (u \cos v,u \sin v,v)\).
   1. Compute \(\vec{X}_u\bigl(\sqrt{2},\pi/4\bigr)\), \(\vec{X}_v\bigl(\sqrt{2},\pi/4\bigr)\), and \(d\vec{X}\bigl(\sqrt{2},\pi/4\bigr)\).
   2. Express the vector \((1,5,2)\) as a linear combination of \(\vec{X}_u\bigl(\sqrt{2},\pi/4\bigr)\) and \(\vec{X}_v\bigl(\sqrt{2},\pi/4\bigr)\).
   3. Find the formula for any space curve \(\vec{x}(t)\) that lies on the helicoid \(S\) for which \(\vec{x}(0) = (1,1,\pi/4)\) and \(\vec{x}\,'(0) = (1,5,2)\).