Consider the surface given in cylindrical coordinates by the equation z = r cos 3θ.
(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 xy-plane by choosing a particular constant value for θ.)
(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.
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