Activity 13.5.5

Suppose that $\vF$ is a continuous path-independent vector field (in $\R^2$ or $\R^3$) on some region $D$.

1. Let $P$ and $Q$ be points in $D$ and let $C_1$ and $C_2$ be oriented curves from $P$ to $Q$. What can you say about $\int_{C_1}\vF\cdot d\vr$ and $\int_{C_2}\vF\cdot d\vr$?

2. Let $C = C_1 - C_2$. Explain why $C$ is a closed curve.

3. Calculate $\oint_C\vF\cdot d\vr$.

4. Write a sentence that summarizes what we can conclude about line integrals of $\vF$ at this point in the activity.

5. Now let us suppose that $\vG$ is a continuous vector field on a region $D$ for which $\oint_C\vG\cdot d\vr = 0$ for all closed curves $C$. Pick two points $P$ and $Q$ in $D$. Let $C_1$ and $C_2$ be oriented curves from $P$ to $Q$. What type of curve is $C = C_1 - C_2$?

6. What is $\oint_C\vG\cdot d\vr$? Why?

7. What does that tell you about the relationship between $\int_{C_1}\vG\cdot d\vr$ and $\int_{C_2}\vG\cdot d\vr$?

8. Explain why this shows that $\vG$ is path-independent.