Consider the surface given in cylindrical coordinates by the equation z = r cos 3. (a)

Chapter 2, Problem 35

(choose chapter or problem)

Consider the surface given in cylindrical coordinates by the equation z = r cos 3. (a) Describe this surface in Cartesian coordinates, that is, as z = f (x, y). (b) Is f continuous at the origin? (Hint: Think cylindrical.) (c) Find expressions for f/x and f/y at points other than (0, 0). Give values for f/x and f/y at (0, 0) by looking at the partial functions of f through (x, 0) and (0, y) and taking one-variable limits. (d) Show that the directional derivative Du f (0, 0) exists for every direction (unit vector) u. (Hint: Think in cylindrical coordinates again and note that you can specify a direction through the origin in the x y-plane by choosing a particular constant value for .) (e) Show directly (by examining the expression for f/y when (x, y) = (0, 0) and also using part (c)) that f/y is not continuous at (0, 0). (f) Sketch the graph of the surface, perhaps using a computer to do so.