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Section 6 Wrap Up Questions

  1. Explain how the graph of a vector field can help you decide whether or not it is conservative.

  2. Are the following statements true or false? Give reasons for your answers.

    1. Any vector field whose component functions are constants is conservative.

    2. If \(\vec{F}\) is conservative and \(C\) is a curve parameterized by \(\vec{r}(t)\) with \(a\leq t\leq b\text{,}\) then \(\int_{C} \vec{F} \cdot d\vec{r} =\vec{r}(b)-\vec{r}(a)\text{.}\)

    3. If \(\vec{F}\) is conservative and \(C\) is a closed curve, then \(\int_{C} \vec{F} \cdot d\vec{r} =0\text{.}\)

    4. If \(\vec{F}\) is conservative and \(C\) is not closed, then \(\int_{C} \vec{F} \cdot d\vec{r} \neq 0\text{.}\)

    5. If \(\vec{F}\) is not conservative and \(C_1\) and \(C_2\) are paths between the same two points, then \(\int_{C_1} \vec{F} \cdot d\vec{r} \neq \int_{C_2} \vec{F} \cdot d\vec{r}\text{.}\)

  3. Explain why the two definitions of path-independent vector field are equivalent.

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