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Page 1: What are the contours of  f(x,y) = x-3y ?

What are the contours of f(x,y) = x-3y?

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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

Page 2: What are the contours of  f(x,y) = x-3y ?

Which of the following contour plots best represents φ(x,y) = 2x + y ?

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1.

2.

None of the above3.

Page 3: What are the contours of  f(x,y) = x-3y ?

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.

Page 4: What are the contours of  f(x,y) = x-3y ?

Which of the following diagrams represents the unit vectors i and j?

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1.

2.

3.

4.

Page 5: What are the contours of  f(x,y) = x-3y ?

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

Page 6: What are the contours of  f(x,y) = x-3y ?

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

Page 7: What are the contours of  f(x,y) = x-3y ?

gradf can also be written as which of the following?

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f.1.

.f2.

f3.

f4.

Page 8: What are the contours of  f(x,y) = x-3y ?

Which of the following shows positive divergence?

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1.

2.

3.

Page 9: What are the contours of  f(x,y) = x-3y ?

F = xi +xyj – xyzkFind divF

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1. 1 + x – xy

2. i + xj – xyk

3. x + y – xy

4. x + xy – xyz

Page 10: What are the contours of  f(x,y) = x-3y ?

v = axyi – ½y²j – yzkFind div v

(a + 2)y

(a – 2)y

(2 – a)y

a – 2y

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1. (a + 2)y

2. (a – 2)y

3. (2 – a)y

4. a – 2y

Page 11: What are the contours of  f(x,y) = x-3y ?

Using div v = (a – 2)y, find a to ensure fluid is incompressible

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1. 0

2. 1

3. 2

4. 3

Page 12: What are the contours of  f(x,y) = x-3y ?

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 + yk

4. - xzi + yzj + yk

Page 13: What are the contours of  f(x,y) = x-3y ?

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 - 4zk

3. -2zi + (4yz+1)xj + (3-2yz)yk

4. -2zi – (4yz+1)xj + (3-2yz)yk

Page 14: What are the contours of  f(x,y) = x-3y ?

The curl of a vector field measures which of the following?

The greate

st r..

.

The rate

at wh...

The rota

tion 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

Page 15: What are the contours of  f(x,y) = x-3y ?

Which of the following statements are true?

div(grad f)

ex...

grad(d

iv v) e

x...

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

Page 16: What are the contours of  f(x,y) = x-3y ?

A = xyzi + xy²j - x²zk and φ = 3xyz²Find ( )φ

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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)

Page 17: What are the contours of  f(x,y) = x-3y ?

φ = 2xy²z³ - x²z² +yz. Find

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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

Page 18: What are the contours of  f(x,y) = x-3y ?

If φ = x²y³z³. Find the rate of change of φ at (1,3,1) in the direction of

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kji 3210

1

5

10241.

5

1032.

5

10643.

10

3364.

Page 19: What are the contours of  f(x,y) = x-3y ?

Given

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1 2

1.,3.C C

drFdrF

1. 2

2. 3

3. 4

4. Don’t Know

C

drF.Find where C is the closed path shown on the diagram below

Page 20: What are the contours of  f(x,y) = x-3y ?

r = 3ti + (8t – 2t²)j. Find

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dt

rd

1. 7t

2. 3ti + 4tj

3. 11 – 4t

4. 3i + (8 – 4t)j

Page 21: What are the contours of  f(x,y) = x-3y ?

A magnetic field B is given by. Find

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zkz ˆˆˆB 3 B

zp

ˆ2ˆ

3

1.

zp

ˆ2ˆ

3

2.

zp

ˆ2ˆ

4

3.

zp

ˆ2ˆ

4

4.

Page 22: What are the contours of  f(x,y) = x-3y ?

Given

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What is divF ? ˆFˆFˆF),,(F rr r

)()sin()sin(sin

1 22

rFFrFrrr r

1.

)()sin()sin(sin

1 22 rrF

rFrFr

r

2.

)()(sin)sin(sin

1

rFFFrrr r

3.

Page 23: What are the contours of  f(x,y) = x-3y ?

Find

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ˆˆcosˆsin 3 rrrrF F.

coscotsin3 2r1.

coscotsin3 2r2.

sin

cossin3

22r

3.

sin

cossin3

22r

4.