Calculus III Final Examination, Fall 1999.

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    Do any 6 questions. Your paper must be postmarked before 12/24/99. Send to my home, via Federal Express or Priority Mail. Avoid snail mail. If you have troubles, email me.

Problem 1. Find the maxima, minima, and saddle points of the function
f(x,y) = (x-y) (x² + x y + y²-1).

Problem 2. Use the method of Lagrange's multipliers to find the maxima, minima, and/or saddle points of the function
f(x,y,z) = (x-1)²+(y-1)²+(z-1)² upon the surface
x²+y²-x = 0. Then determine whether each of these points are local maxima or minima or saddle points compared with neighboring points on the surface. You may parametrize the surface with r and q .

Problem 3. Given a point A(-2, -1, 1) and the plane P: x-2y+2z+1=0
and the line L: (x,y,z) = (-1,2,1) + t(2,1,-1).
(a) Find a point on L whose distance from A is 3 times its distance from P. (2 answers).
(b) Find a line going through A and parallel to P and intersecting L at some point Q. Find the coordinates of Q also. (Warning, answers may contain irrational numbers.)

Problem 4.Find the volume of the space inside the cone z = Ö(x²+y²)
and inside the sphere x²+y²+z²= 2x. [Hint: their intersection curve lies on some general cylinder whose equation can be obtained by eliminating z from the two equations, why?]

Problem 5. Verify Green's Theorem for ò_C (y/r²) dx - (x/r²) dy,
where C is the circle r = 1. Here x = r cos q and y = r sin q . Explain why Green's theorem is not true in this case.
[Hint: On the curve C, express x and y as functions of some parameter, such as q , then replace dx by x' dt, dy by y' dt, etc.]

Problem 6. Transform Laplace's equation f_xx + f_yy + f_zz = 0 for a scalar field f(x,y,z) into cylindrical coordinates.
[Hint: From x = r cosq , y = r sinq , etc., get
f_x = f_r r_x + f_q q _x + f_z z_x . . . (i).
Of course the last term is 0. Next do f_xx by deriving (i) relative to x,
but you have to transform each f_{r x} into f_rr r_x + f_rq q _x ,
and similarly for f_{q x} . After you got f_xx , do the same for f_yy , but f_zz needs no manipulation. Then substiture into the left side of Laplace's equation.]

Problem 7. Let S be the surface z = f(x,y) = Ö(x²+1-y²) .
Find the slopes of the surface at the point A(1,1,1) along the northeast, east, and southeast directions, i.e., dx:dy = 1:1, 1:0 or 1:-1 . Let g(x,y,z) = z - f(x,y). Find Ñg. At the point A, use it as a normal to find the equation of the tangent plane T. Prove T and S possess some line in common.

Problem 8. Let a vector field be defined by F = (x r, y r, z r), where r = Ö(x²+y²+z²).
Find its curl and divergence. Find the triple integral of Ñ·F over the interior of the unit sphere x²+y²+z² = 1.