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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).