The statement provided is:

Let $\vec{F}$ be a continuous vector field, and let $-C$ be the same curve as $C$, but with the opposite orientation. Then $\int_{-C} \vec{F} \cdot d\vec{r} = - \int_{C} \vec{F} \cdot d\vec{r}.$

To determine whether this is true or false, let’s examine what reversing the orientation of the curve does to the line integral.

Explanation:

1. The line integral $\int_{C} \vec{F} \cdot d\vec{r}$ represents the work done by the vector field $\vec{F}$ along the path $C$ in the specified orientation.

2. When the orientation of the curve $C$ is reversed (resulting in $-C$), the differential element $d\vec{r}$ also changes its direction. Specifically, $d\vec{r}$ along $-C$ is equal to $-d\vec{r}$ along $C$.

3. Substituting $-d\vec{r}$ in the integral over $-C$ gives: $\int_{-C} \vec{F} \cdot d\vec{r} = \int_{C} \vec{F} \cdot (-d\vec{r}) = - \int_{C} \vec{F} \cdot d\vec{r}.$

Conclusion:

The statement is true. Reversing the orientation of the curve changes the sign of the line integral.

Would you like further details on this explanation?

Here are some related questions to expand your understanding:

1. What happens to a line integral if the vector field $\vec{F}$ is conservative?
2. How would the answer change if $C$ were a closed curve?
3. What are the physical interpretations of line integrals in vector fields?
4. How does the Fundamental Theorem for Line Integrals relate to path orientation?
5. How do line integrals apply to work done in physics?

Tip: When working with line integrals, remember that the orientation of the curve is essential—it directly affects the sign of the integral.