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1.  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.

2.  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.

3.  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.  \(30\)