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Which of the following contour plots best represents φ(x,y) = 2x + y ? 1. 1 2 3 2. None of the above 3.
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What are the contours of f(x,y) = x-3y?
1 2 3 4
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1. Straight lines of gradient -3
2. Straight lines of gradient 3
3. Straight lines of gradient -1/3
4. Straight lines of gradient 1/3
Which of the following contour plots best represents φ(x,y) = 2x + y ?
1 2 3
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1.
2.
None of the above3.
If C is a closed contour, what symbol is used when evaluating a
line integral?
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C
1.
2.
C
3.
4.
Which of the following diagrams represents the unit vectors i and j?
1 2 3 4
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1.
2.
3.
4.
φ(x,y,z)= xy - x²yz² + y²zFind gradφ
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1. xyi - x²yz²j + y²zk
2. yi - x²z²j – 2x²y²zk
3. (y – 2yz²)i + (x - x²z² + 2z)j + y²k
4. (y – 2xyz²)i + (x - x²z² + 2yz)j + (y²-2x²yz)k
φ = 3x + 2xy²z - y²z²Find gradφ
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1. (3+2y²z)i + (2x–z)2yzj +(x–z)2y²k
2. (3+2y²z)xi + (2x–z)2y²zj +(x–z)2y²zk
3. 3i + 4xyzj – 2y²zk
4. 3i + 4xyzj + 2y²zk
gradf can also be written as which of the following?
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f.1.
.f2.
f3.
f4.
Which of the following shows positive divergence?
1 2 3
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1.
2.
3.
F = xi +xyj – xyzkFind divF
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1. 1 + x – xy2. i + xj – xyk3. x + y – xy4. x + xy – xyz
v = axyi – ½y²j – yzkFind div v
(a + 2)y
(a – 2)y
(2 – a)y
a – 2y
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1. (a + 2)y2. (a – 2)y3. (2 – a)y4. a – 2y
Using div v = (a – 2)y, find a to ensure fluid is incompressible
0 1 2 3
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1. 02. 13. 24. 3
F = xi + xyj – xyzk Find curlF
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1. (-z-y)xyi + (yz+1)xj +(y-1)xk
2. (-z-y)xyi - (yz+1)xj +(y-1)xk
3. -xzi – yzj + yk4. - xzi + yzj + yk
F = 3yzi – xzj + 2xy²zk Find curlF
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1. (4yz+1)xi + (2yz-3)yj - 4zk
2. (4yz+1)xi - (2yz-3)yj - 4zk3. -2zi + (4yz+1)xj + (3-
2yz)yk4. -2zi – (4yz+1)xj + (3-
2yz)yk
The curl of a vector field measures which of the following?
The greatest
r...
The rate at w
h...
The rotation a...
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1. The greatest rate of change at any point
2. The rate at which fluid is flowing away from a point
3. The rotation at a point
Which of the following statements are true?
div(grad f)
ex...
grad(div
v) ex...
div(curl v
) ex...
curl(
div v) ex...
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1. div(grad f) exists and is a scalar
2. grad(div v) exists and is a scalar
3. div(curl v) exists and is a vector
4. curl(div v) exists and is a vector
A = xyzi + xy²j - x²zk and φ = 3xyz²Find ( )φ
1 2 3 4
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.A
1. 3xyz²(yzi + xyj – 2x²k)
2. 3xyz²(yz + xy – 2x²)3. 3xyz²(yzi + 2xyj –
x²k)4. 3xyz²(yz + 2xy – x²)
φ = 2xy²z³ - x²z² +yz. Find
1 2 3 4
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2
1. z(z + 4xz² +12xy²)
2. z²i + 4xz³j + 12xy²zk
3. z²(2y + x + 4xyz + 6xy²z)
4. -2z² + 4xz³ + 12xy² - 2x²
If φ = x²y³z³. Find the rate of change of φ at (1,3,1) in the direction of
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kji 32101
51024
1.
5103
2.
51064
3.
103364.
Given
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1 2
1.,3.C C
drFdrF
1. 22. 33. 44. Don’t Know
C
drF.Find where C is the closed path shown on the diagram below
r = 3ti + (8t – 2t²)j. Find
1 2 3 4
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dtrd
1. 7t2. 3ti + 4tj3. 11 – 4t4. 3i + (8 – 4t)j
A magnetic field B is given by. Find
1 2 3 4
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zkz ˆˆˆB 3 B
zp
ˆ2ˆ3
1.
zp
ˆ2ˆ3
2.
zp
ˆ2ˆ4
3.
zp
ˆ2ˆ4
4.
Given
1 2 3
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What is divF ? ˆFˆFˆF),,(F rr r
)()sin()sin(
sin1 2
2
rFFrFr
rr r
1.
)()sin()sin(
sin1 2
2 rrFr
FrFrr
2.
)()(sin)sin(
sin1
rFFFr
rr r
3.
Find
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ˆˆcosˆsin 3 rrrrF F.
coscotsin3 2r1.
coscotsin3 2r2.
sincossin3
22r
3.
sincossin3
22r
4.