21a Final Exam Answers (Spring 97)

1) A, B, D
2) B, C, E
3) C
4) B
5) I, A, H, J, B, D, K, G, F, G, F, E
6) A
7) C
8) B
9) E
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1) C(x,y,z)=5x+2y+z=900

grad C is never 0, so we need (grad u) = q (grad C) =>

1/3 x^(-2/3) y^(1/4)  z^(2/9) = 5q (I)   => u=15qx
1/9 x^(1/3)  y^(-8/9) z^(2/9) = 2q (II)  => u=18qy
2/9 x^(1/3)  y^(1/9)  z^(-7/9) = q (III) => u=4.5qz

q=0 implies u=0, what is no good. Then x=1.2y and z=4y =>
=> 6y+2y+4y=900 => y=75; x=90; z=300 is the solution.

2) Front: S // to yz-plane, so n=-i => F.n=2 => Flux = Int 2 dsigma =
      2 Area = 2.5.3 = 30
   Top: S is on 2z+x=8, so it is the graph of f(x,y)=4-x/2 =>
     Flux = Int (F.n) dsigma =
     = Int (0 to 5) (0 to 2) (-2i+3j-(4-x/2+1)k).(-1/2i-0j-k) dx dy =
     = Int (0 to 5) (0 to 2) (6-x/2) dx dy = 55

	TOTAL FLUX = 85

3)
a) F is conservative; potential f(x,y)=xe^y+(y+1)e^x
b) Line integral = f(r(Pi))-f(r(0))=f(0,-1)-f(0,0)=0-e=-e

4)
a) Line integral around any closed curve = 0 <==> curl F = 0
   curl F = (-bx/6-ax)i-(bz-by/6+6z-y)j+(az-z)k
   So a=1, b=-6.

b) Flux over closed surface = 0 <==> div F = 0
   div F = 4+2x+2ay+bx-4
   So a=0, b=-2.

5)
a) Area = Int_(shadow) sqrt(1+16x^2+16y^2) dx dy = 
        = Int (0 to 2Pi) (0 to 1) sqrt(1+16r^2) r dr dtheta =
        = Pi/24(17^1.5-1)
b) Volume
   = Int (0 to 2Pi) (1 to 2) (0 to sqrt(z/2)) 1 r dr dz dtheta =
   = Int (0 to 2Pi) (1 to 2) z/4 dz dtheta = 
   = Int (0 to 2Pi) 3/8 dtheta = 3Pi/4