1. What does the line integral $W = \int_C \mathbf{F} \cdot d\mathbf{r}$ compute when $\mathbf{F}$ represents a force field and $C$ is a smooth curve?
- A.The flux of $\mathbf{F}$ across the curve $C$
- B.The work done by $\mathbf{F}$ on a particle moving along $C$
- C.The arc length of the curve $C$ weighted by $\|\mathbf{F}\|$
- D.The circulation of $\mathbf{F}$ around the boundary of a surface
View Answer
Answer: The work done by $\mathbf{F}$ on a particle moving along $C$
Recall the definition of work: In vector calculus, the work done by a force field $\mathbf{F}$ on a particle moving along a curve $C$ is defined as $W = \int_C \mathbf{F} \cdot d\mathbf{r}$. The dot product projects the force onto the direction of motion at each point. Why the correct answer works: Option B directly states the physical interpretation of this integral: it computes the work done by the force along the path. Why distractors fail: Option A describes flux, which uses $\int_C \mathbf{F} \cdot \mathbf{n}\, ds$ in 2D. Option C describes a scalar line integral $\int_C \|\mathbf{F}\|\, ds$, not a vector line integral. Option D describes circulation, which specifically requires a closed curve; the general work integral does not require $C$ to be closed.