Let C be the unit circle $x ^ { 2 } + y ^ { 2 } = 1$ oriented in a counter-clockwise direction.Say whether the following statements are true or false.If $\int_{c} \vec{F} \cdot d \vec{r} \neq 0$ , we can conclude that $\vec { F }$ is not path-independent field.

Question

Quizplus · Accepted Answer

If the line integral of a vector field $\vec{F}$ around a closed path (like the unit circle C) is non-zero, it indicates that the work done by $\vec{F}$ in moving a particle around the path depends on the path taken. This is a characteristic of a path-dependent field, meaning $\vec{F}$ cannot be path-independent.

Let C Be the Unit Circle x2+y2=1x ^ { 2 } + y ^ { 2 } = 1x2+y2=1

Let C Be the Unit Circle $x ^ { 2 } + y ^ { 2 } = 1$