If C₁ and C₂ are two curves with the same starting and ending points, then $\int _ { c _ { 1 } } \vec { F } \cdot \overline { d r } = \int _ { c _ { 2 } } \vec { F } \cdot \overline { d r }$ , for any vector field $\vec{F}$

Question

Quizplus · Accepted Answer

The equality of the two integrals depends on whether the vector field $\vec{F}$ is conservative or not. If $\vec{F}$ is conservative, then the path integral between any two points is independent of the path taken, and the statement would be true. However, without specifying that $\vec{F}$ is conservative, we cannot guarantee the equality for any vector field $\vec{F}$.

If C1 and C2 Are Two Curves with the Same ∫c1F⃗⋅dr‾=∫c2F⃗⋅dr‾\int _ { c _ { 1 } } \vec { F } \cdot \overline { d r } = \int _ { c _ { 2 } } \vec { F } \cdot \overline { d r }∫c1​​F⋅dr=∫c2​​F⋅dr

If C₁ and C₂ Are Two Curves with the Same $\int _ { c _ { 1 } } \vec { F } \cdot \overline { d r } = \int _ { c _ { 2 } } \vec { F } \cdot \overline { d r }$