Subtopic: Physical interpretations – line, surface and volume integrals
Question Statement:
A force field $\mathbf{F}$ is conservative in a simply connected domain $D$ if and only if:
Options:
(A) $\oint_C \mathbf{F} \cdot d\mathbf{r} = 0$ around every simple closed curve $C$ in $D$
(B) $\nabla \cdot \mathbf{F} = 0$ throughout $D$
(C) $\mathbf{F} \cdot \mathbf{r} = 0$ for all position vectors $\mathbf{r}$
(D) The work done depends strictly on the path taken between two points
Answer Key:
A
Step-by-Step Solution & Explanation
A force field is conservative if and only if the work done between any two points is independent of the path connecting them. This is mathematically equivalent to the circulation around any simple closed curve being zero: $\oint_C \mathbf{F} \cdot d\mathbf{r} = 0$.