Due May 26, 2020
Math 281 - Test 4
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1. Let F( x, y, z ) = e x sin yi + e x cos yj + xyzk .
a. Find curl F.
b.
Find
div F.
2. Let C be the curve defined by
a. Find
# 3z
b. Find
#
C
C
2
x = 3 cos t, y = 3 sin t, z = 4t, 0 ! t ! " .
ds .
$2 y dx + 2 x dy + ( x 2 + y 2 ) dz .
Due May 26, 2020
Math 281 - Test 4
page 2 of 4
3. Determine whether or not the vector field F is conservative. If it is conservative, find a
function f such that F = %f . If F is not conservative, explain why not.
a. F( x, y) = 3e3 x sin y i + (3 + e3 x cos y) j
b.
F( x, y) = (e y + y) i + ( xe y $ x ) j
4. Use Green’s Theorem to find # (3 ye3 x + xy) dx + (e3 x + x 2 ) dy where C is the path of line
C
segments from (0, 0) to (2, –2) to (2, 2) back to (0, 0).
Due May 26, 2020
Math 281 - Test 4
page 3 of 4
5. Let F( x, y, z ) = ( y + z )i + ( x + z ) j + ( x + y)k . [One can test that F is conservative. Assume this.]
a. Given that F is conservative, find a function f such that %f = F .
b. Evaluate the integral # F & dr where C is the path of line segments from (2, 1, 2) to (2, 1, 5)
C
to (2, 7, 5). Use a method of your choice. [The Fundamental Theorem of Line Integrals will be
easiest if you trust your work in part a. Note that C is not a closed curve.]
Due May 26, 2020
Math 281 - Test 4
page 4 of 4
6. You are asked to calculate the line integral '
) F & dr where C is the ellipse produced by
(C
intersecting the plane x + z = 0 with the cylinder x 2 + y 2 = 1, and F( x, y, z ) = z, x, y . Because
''
you see that curl F = 1, 1, 1 , you choose to use Stokes’ Theorem '
) F & dr = )) curl F & dS and
(C
((S
instead to calculate the surface integral, where S is the portion of the plane paramaterized by
r(u, v) = u cos v, u sin v, $ u cos v , 0 ! u ! 1,
0 ! v ! 2"
To that end you do the following:
a.
Calculate ru (u, v) .
b.
Calculate rv (u, v).
c.
Calculate ru # rv .
2"
'' curl F & dS = '
)
))
(0
((S
1
'
) 1,1,1 & $ru # rv % du dv .
(0
d.
Calculate
[e.
Optional:
C is paramaterized by r(t ) = cos t, sin t, $ cos t ,
0 ! t ! 2".
Show '
) F & dr calculated directly gives you the same answer. For modest extra credit.]
(C
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