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1.  If \(\vec{F} = \langle y, z, x \rangle\), compute \(\vec{curl}\vec{F}\)

a.  \((1,1,1)\)

b.  \((-1,1,1)\)

c.  \((1,-1,1)\)

d.  \((1,1,-1)\)

e.  \((-1, -1,-1)\)

For each of the following, determine whether or not the vector field is conservative.

2.  \(\vec{F}(x,y) = \langle xy, x^2 \rangle\).

a.  It is conservative.

b.  It is not conservative.

3.  \(\vec{F}(x,y) = \langle 2xy, x^2 \rangle\).

a.  It is conservative.

b.  It is not conservative.

4.  \(\vec{F}(x,y) = \langle ye^{xy}, xe^{xy} \rangle\).

a.  It is conservative.

b.  It is not conservative.

5.  \(\vec{F}(x,y) = \langle \sin(y), x\cos(y)\rangle\).

a.  It is conservative.

b.  It is not conservative.

6.  Say the vector field \(\vec{F} = \langle \sin x, 0\rangle\) describes the velocity field of a fluid. Then the fluid

a.  is getting less dense everywhere.

b.  is getting more dense everywhere.

c.  is getting less dense in regions that look like parallel vertical stripes, where each stripe extends up and down to infinity.

d.  is getting less dense in regions that look like parallel horizontal stripes, where each stripe extends left and right to infinity.

e.  Is getting less dense in regions that look like sine waves.

7.  Could the fluid in question 6 be liquid water?

a.  No, because liquid water cannot be compressed.

b.  No, because liquid water cannot flow in swirls.

c.  Yes

8.  Evaluate \(\int_C x^2+y^2 ds\), where \(C:\vec{r}(t) = \langle \cos t, \sin t \rangle\), \(0 \leq t \leq 2\pi\) (If you think about this, you can save yourself some work).

a.  0

b.  1

c.  2

d.  \(\pi\)

e.  \(2\pi\)

9.  Evaluate \(\int_C \langle e^x, xy \rangle \cdot d\vec{r}\), where \(C:\vec{r}(t) = \langle t^2, t \rangle\), \(0 \leq t \leq 1\)

a.  \(e + \frac{1}{4}\)

b.  \(e^2\)

c.  \(e - \frac{3}{4}\)

d.  \(e^2 + \frac{1}{4}\)

e.  \(1\)

10.  Say \(C_1:\vec{r}_1(t), a \leq t \leq b\), and \(C_2:\vec{r}_2(t), c \leq t \leq d\) are two piecewise smooth curves such that \(\vec{r}_2(c) = \vec{r}_1(a)\), and \(\vec{r}_2(d) = \vec{r}_1(b)\). Is it true that $$\int_{C_2} \vec{F} \cdot d\vec{r} = \int_{C_2} \vec{F} \cdot d\vec{r}?$$

a.  Definitely yes.

b.  Definitely no.

a.  It could possibly happen.

11.  Say \(C_1:\vec{r}_1(t), a \leq t \leq b\), and \(C_2:\vec{r}_2(t), c \leq t \leq d\) are two piecewise smooth curves such that \(\vec{r}_2(c) = \vec{r}_1(a)\), and \(\vec{r}_2(d) = \vec{r}_1(b)\), and \(\vec{F}\) is conservative. Is it true that $$\int_{C_2} \vec{F} \cdot d\vec{r} = \int_{C_2} \vec{F} \cdot d\vec{r}?$$

a.  Definitely yes.

b.  Definitely no.

a.  It could possibly happen.

12.  Evaluate \(\int_C \langle x+y, x-y \rangle \cdot d\vec{r}\), where \(C\) is the path that starts at (0,0), goes in a straight line to (1,1), then goes in a straight line to (3,-1), then goes in a straight line to (5, 1), and then goes in a straight line to (6,0). (Hint: if you're working hard on this one, you're working too hard.)

a.  \(12\)

b.  \(16\)

a.  \(18\)

b.  \(24\)

b.  \(30\)

13.  Evaluate \(\int_C x^2+y^2 ds\), where \(C:\vec{r}(t) = \langle \cos t, \sin t \rangle\), \(0 \leq t \leq 2\pi\) (If you think about this, you can save yourself some work).

a.  0

b.  1

c.  2

d.  \(\pi\)

e.  \(2\pi\)

14.  Evaluate \(\int_C \langle e^x, xy \rangle \cdot d\vec{r}\), where \(C:\vec{r}(t) = \langle t^2, t \rangle\), \(0 \leq t \leq 1\)

a.  \(e + \frac{1}{4}\)

b.  \(e^2\)

c.  \(e - \frac{3}{4}\)

d.  \(e^2 + \frac{1}{4}\)

e.  \(1\)

15.  Evaluate \(\int_C \langle -y, x\rangle \cdot d\vec{r}\), where \(C:\vec{r}(t) = \langle \cos t, \sin t \rangle\), \(0 \leq t \leq 2\pi\) (If you think about this, you can save yourself some work).

a.  \(-4\pi\)

b.  \(-2\pi\)

c.  0

d.  \(2\pi\)

e.  \(4\pi\)

16.  Describe the following surface \(S: \vec{r}(u,v) = \langle v\cos u, v\sin u, v\rangle\) where \(0 \leq u \leq 2\pi\), and \(0\leq v\leq 1\).

a.  A finite circular cylinder

b.  An infinite circular cylinder

c.  A finite cone

d.  An infinite cone

e.  A hemisphere