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April 24th, 2012, 12:17 AM   #1
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Define a function z = f(x,y) by f(0,0) = 0 and otherwise

Define a function z = f(x,y) by f(0,0) = 0 and otherwise.

f(x,y) =(x^2 y) / (x^2+y^2 )

a. Show that in polar coordinates this function may be expressed (for r? 0) as z = r ?cos?^2 (?)sin(?)

b. Show that if ? is fixed then the graph is given by z = mr, a line of slope
m= ?cos?^2 (?)sin?(?).
(Note that this says that the surface z = f(x,y) is what is called a ruled surface.)

c. Compute the directional derivatives of z in the ? direction. Does Df exist at the point (0,0)? Explain.
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