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MATH 227
Fall 2008Test 4
1.(20pts)(a) Show that F(x,y,z) = (y2+2x)i + (2xy+z2+2y)j +(2yz+2z)k
is a conservative vector field in R3 by computing curl(F).
curl(F) = det
?
?
i j k
?x ?y ?z
y2 + 2x 2xy + z2 + 2y 2yz + 2z
?
? =
< 2z ?2z,?(0?0),2y ?2y > =< 0,0,0 >
(b) Find a potential function, f , for F.
fx(x,y,z) = y2 + 2x , so f(x,y,z) = xy2 + x2 + C(y,z). Now
?
?y(xy
2 + x2 + C(y,z)) = 2xy + z2 + 2y =? 2xy + ?
?yC(y,z) = 2xy + z
2 + 2y.
Thus C(y,z) = yz2 + y2 + D(z). So ??z(xy2 + x2 + yz2 + y2 + D(z)) =
2yz + 2z =? ddzD(z) = 2z and D(z) = z2 + d, where d is a constant.
We conclude that ?f(x,y,z) = F(x,y,z),
where f(x,y,z) = xy2 + x2 + yz2 + y2 + z2 + d
(c) Find the work done by F on a particle which moves on any curve
joining (0,0,0) to (1,1,1).
The work done by F along C, where C is any pws curve joining (0,0,0)
to (1,1,1) is integraltext
C
F·dr = integraltext
C
?f ·dr = f(1,1,1)?f(0,0,0) = 5.
2.(20pts) A region ? is bounded by the curves y = x and y = x2.
Consider the vector f...

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