If $\int _ { C } \bar { F } \cdot \overline { d r } = 0$ , then $\vec{F}$ is perpendicular to the curve C at every point.

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The integral $\int _ { C } \bar { F } \cdot \overline { d r } = 0$ implies that the work done by the force field $\bar{F}$ along the curve $C$ is zero, but it does not necessarily mean that $\vec{F}$ is perpendicular to the curve $C$ at every point. The force could be parallel and in the opposite direction to the displacement in some segments, or the force and displacement could be perpendicular in some parts and not in others, as long as the total work done is zero.

If ∫CFˉ⋅dr‾=0\int _ { C } \bar { F } \cdot \overline { d r } = 0∫C​Fˉ⋅dr=0

If $\int _ { C } \bar { F } \cdot \overline { d r } = 0$