The unit normal is given by which of the following?

1 2 3 4

0% 0%0%0%

1.

2.

3.

4.

Find a unit normal to the plane 4x + y – 2z = 3

1 2 3 4

0% 0%0%0%

kji 2413

1

1.

kji 2421

1

2.

kzjyix 2413

1

3.

kzjyix 2421

1

4.

Find a unit vector normal to the sphere (x + 3)² + (y – 1)² + 2z² = 5 at (0, 0, 1).

1 2 3 4

0% 0%0%0%

kji 42628

14

1.

kji 42612

3

2.

kji 2314

14

3.

kji 236

3

4.

Which of the following vector fields are conservative?

1 2 3 4

0% 0%0%0%

1. F=xyi + 2y²zj + 3xyzk

2. F=y²i + 2xyj + 2zk

3. F=2x²zi + xyzj - 2xyk

4. F=2yzi + 2xzj + 2xyk

Which of the following statements are true?

1 2 3 4

0% 0%0%0%

B

A

drF.1. only depends on the end points of A and B

2. for all C

3. for a conservative field F

4. All gradient fields are conservative

C

drF 0.

0 F

represents:

1 2 3 4

0% 0%0%0%

C

dSyxF ),(

1. The area beneath the surface z=F(x,y) but above the curve C.

2. The area beneath the surface z=F(x,y) and below the curve C.

3. The area above the surface z=F(x,y) and above the curve C.

4. The area above the surface z=F(x,y) but below the curve C.

Which of the following statements is true?

1 2 3 4

0% 0%0%0%

S

dSrF ).(

S

dSrF )(

1. evaluates to a scalar

2. evaluates to a scalar

3. Both of the above evaluate to scalars

4. Neither of the above evaluate to scalars

Find where C is the curve y=x²+2 starting from x=0, y=0 and

ending at x=1, y=1

1 2 3 4

0% 0%0%0%

1. ½ + y

2. 1 + 2y

3. 3

4. 7

C

dxyx )21(

Find where, on C, x and y are given in terms of the parameter t by x=2t and y=t²+1 for t varying from

0 to 1.

1 2 3

0% 0%0%

1. 7/3

2. 14/3

3. ½ + y

C

dxyx

In general

1 2 3

0% 0%0%

C C C

dxyxfdyyxfdsyxf ),(),(),(

1. True

2. False

3. Don’t Know

F = xyi + y²jFind from (0,0) to (1,3) where

C is the curve y = 3x

1 2 3 4

0% 0%0%0%

C

dr.F

1. 10

2. 270

3. ½y + 9

4. (9/2)y + 1

F = 3xyzi + x²yj – 2xyz²kC is a curve from A=(0,0,0) to B=(1,1,1) given by x=y=z=t, 0≤t≤1. Find .

1 2 3 4

0% 0%0%0%

C

drF.

1. 3/10

2. 3/5

3. i/4 + j/4– k/5

4. 3i/4 + j/4 – 2k/5

Evaluate where C represents the contour y=x²+1 from (0,1) to (1,2)

1 2 3 4

0% 0%0%0%

C

dryx2

ji6

1

15

8

1.

ji6

5

15

8

2.

ji2

3

3

1

3.

ji6

23

3

1

4.

Find where F = 2x²i + xy²j + xzk and C is the curve

y = x², z = x³ from (0,0,0) to (1,1,1)

1 2 3 4

0% 0%0%0%

C

drF.

kji4

21

3

14

2

7

1.

kji4

15

3

10

2

5

2.

kji 33

42

3.

kji2

943

4.

Evaluate where A represents the surface of the unit cube 0≤x≤1, 0≤y≤1, 0≤z≤1 and r = xi + 2yj + 3zk

1 2 3 4

0% 0%0%0%

1. 15

2. 6

3. 3

4. 0

A

dSr.

When an electric current flows at a constant rate through a conductor, then

the current continuity equation states that .

1 2 3

0% 0%0%

1. True

2. False

3. Don’t Know

S

dSJ 0.

Evaluate where F = 2xyi + xy²zj + z²k and S is the

surface of the unit cube 0≤x≤1, 0≤y≤1, 0≤z≤1

1 2 3 4

0% 0%0%0%

S

dSF.

12

131.

12

172.

2

53.

4

94.

F = y²i + 3xyjEvaluate where V is the

volume under the plane z=x+y+1 and above z=0 for -1≤x≤2, -1≤y≤2

1 2 3 4

0% 0%0%0%

dVFV

kyx 12

9

1.

ky

4

27

2

92.

k4

633.

k4

274.

Which of the following is Stokes’ Theorem?

1 2 3 4

0% 0%0%0%

dSFdrFC S

.. 1.

dSFdrFSC

.. 2.

dSFdrFC S .3.

dSFdrFSC .

4.

Which of the following can be obtained from Gauss’ Law?

1 2 3 4

0% 0%0%0%

S

QdSE.1.

S

dSE 0. 2.

0. S

QdSE3.

S

QdSE

0

.

4.

Evaluate

1 2 3 4

0% 0%0%0%

C

dyyxdxyx 24132 222

around the rectangle 0≤x≤4, 0≤y≤2 using Green’s Theorem

1. 0

2. 8

3. -2

4. None of the above