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Section 7 Wrap Up Questions

  1. If \(C_1\) and \(C_2\) are different oriented curves from the point \((a,b)\) to the point \((c,d)\text{,}\) do you expect the integrals \(\int_{C_1} \vec{F}\cdot d\vec{r}\) and \(\int_{C_2} \vec{F}\cdot d\vec{r}\) for a vector field \(\vec{F}\) to be equal?

  2. Answer the following as True or False. Make sure to explain:

    1. If \(\int_{C} \vec{F}\cdot d\vec{r}=0\) then \(C\) is perpendicular to \(\vec{F}\text{.}\)
    2. If \(\int_{C_1} \vec{F}\cdot d\vec{r}>\int_{C_2} \vec{F}\cdot d\vec{r}\) then the curve \(C_1\) is longer than the curve \(C_2\text{.}\)
    3. If \(\int_{C} \vec{F}\cdot d\vec{r}=\int_{C} \vec{G}\cdot d\vec{r}\) then \(\vec{F} =\vec{G}\text{.}\)
    4. If \(\int_{C_1} \vec{F}\cdot d\vec{r}=\int_{C_2} \vec{F}\cdot d\vec{r}\) then \(C_1\) and \(C_2\) are the same curve.
  3. How does the line integral of a scalar-valued function differ from that of a vector-valued function?
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