Homework 5

Homework 5
Math 352, Fall 2011
Instructions: Solve both of these problems. Your solutions must be written in LATEX.
Due Date: Friday, October 28
1. Let S be the paraboloid z = x2 + y 2 , and let σ : R2 → S be the surface patch
σ(x, y) = (x, y, x2 + y 2 ).
Let p = (x, y, z) be a point on S other than (0, 0, 0), and let t be the vector (x, y, 2z).
(a) Find the tangent vectors σx and σy at p.
(b) Find a formula for the unit normal vector Nσ at p.
(c) Find a curve γ on S so that γ(0) = p and γ̇(0) = t.
(d) It follows from part (c) that t is tangent to S. Find values λ and µ so that
t = λσx + µσy .
(e) Find a unit vector v ∈ Tp S so that v is orthogonal to t and t × v = ktk Nσ .
(f) Find a curve γ on S so that γ(0) = p and γ̇(0) = v, where v is the vector you
found in part (e).
2. For each of the following maps, compute the derivative, and determine whether or not
the map is (i) orientation-preserving (ii) equiareal (iii) conformal, and (iv) isometric.
(a) The function f : R2 → R2 defined by f(x, y) = (ex cos y, ex sin y).
(b) The function g : R × (0, ∞) → R2 defined by g(x, y) = x/y, 21 y 2 .
(c) The function h : R2 → R2 defined by h(x, y) = (y + 2, x + 3).