Let C₁be the rectangular loop consisting of four line segments: from (0, 0)to (1, 0), then to (1, 2), then to (0, 2), then back to (0, 0).Suppose C₂ is the triangular loop joining (0, 0)to (1, 0), then to (1, 2)then back to (0, 0), and C₃ is another triangular loop joining (0, 0)to (1, 2), then to (0, 2)and then back to (0, 0). Is it true that $\int _ { c _ { 2 } } \vec { F } \cdot \overline { d r } = \int _ { C _ { 3 } } \vec { F } \cdot \overline { d r } + \int _ { C _ { 1 } } \vec { F } \cdot \overline { d r }$ for any vector field $\vec{F}$ defined on the xy-plane? A)Not possible to decide B)Yes C)No

Question

Quizplus · Accepted Answer

The given equation suggests that the line integral over the triangular loop $C_2$ is equal to the sum of the line integrals over another triangular loop $C_3$ and the rectangular loop $C_1$. However, this relationship does not hold for any vector field $\vec{F}$ because the path taken in the line integral affects the result, and the paths of $C_1$, $C_2$, and $C_3$ are not equivalent in a way that would satisfy this equation for every possible vector field.

Let C1 be the Rectangular Loop Consisting of Four Line

Let C₁be the Rectangular Loop Consisting of Four Line