If $\operatorname { cur } 1 { \vec { F} } = \vec { 0 }$ then by Stokes' Theorem the line integral $\dot { \mathrm { Q } } { \bar { F } \times \vec { d r } }$ is equal to zero, where C is the curve y = x², for 0 $\le$ x $\le$ 2.

Question

Quizplus · Accepted Answer

Stokes' Theorem relates the surface integral of the curl of a vector field over a surface to the line integral of the vector field over the boundary curve of the surface. If the curl of $\vec{F}$ is zero everywhere in a region, then the line integral of $\vec{F}$ around any closed curve in that region is zero. However, the given integral $\dot { \mathrm { Q } }  { \bar { F } \times \vec { d r } }$ seems to be incorrectly formatted, making it unclear what is being asked. Regardless, the principle that a zero curl implies a zero line integral around a closed loop, as per Stokes' Theorem, holds under the assumption of a correctly stated problem.

If cur⁡1F⃗=0⃗\operatorname { cur } 1 { \vec { F} } = \vec { 0 }cur1F=0

If $\operatorname { cur } 1 { \vec { F} } = \vec { 0 }$