Answer Happy

MA330 Homework #6 1. Module 5B: Suppose a > 0 is constant and consider the parametric surface o given by r(0,0) = a sin(o) cos() i+ a sin(o) sin(0)j + a cos(o)k, 0≤0<2m, 0≤ ≤n. (a) Directly verify algebraically that r parameterizes the sphere of radius a > 0 with equation x² + y² +₂²=a², by substituting x = a sin(o) cos(8), y = a sin(o) sin(0), and z = a cos(o) into the left- hand side of this equation, simplifying, and confirming that it agrees with the right- hand side of the equation. (b) Find ro, re, ro x rg. Do this carefully, making sure to keep your and distinct. (c) ro x r, is normal to the sphere. Does it point outward, away from the origin, or does it point inward, towards the origin? Explain with a picture. (d) Verify that ||rx re|| = a² sin(o) as we claimed earlier. (e) Compute the surface area of the sphere [[ ds = [1 ds using Change of Variables for surface integrals and confirming that it is 47a², as expected. (f) Suppose a thin metal surface is shaped into a portion of the sphere of radius a > 0 in the first octant where x, y, z are all non-negative. The density of the surface varies according to the function 5(x, y, z) = xyz grams per square centimeter. Find the mass of the surface. The resulting integral should be accesible doing some simple u-substitutions.