Let $\vec { F } = 4 z \vec { i } - 4 x \vec { k }$ Let C be the circle of radius a parameterized by x = a cos t, y = 0, z = a sin t, 0 $\le$ t $\le$ 2 $\pi$ and let S be the disk in the xz-plane enclosed by C, oriented in the positive y-direction.
(a)Evaluate directly $\int _ { C } \vec { F } \cdot d \vec { r }$ (b)Evaluate directly $\int _ { S } \operatorname { curl } { \vec { F } } \cdot d \vec { A }$ (c)Do these results contradict Stokes' Theorem?

Question

Let $\vec { F } = 4 z \vec { i } - 4 x \vec { k }$ Let C be the circle of radius a parameterized by x = a cos t, y = 0, z = a sin t, 0 $\le$ t $\le$ 2 $\pi$ and let S be the disk in the xz-plane enclosed by C, oriented in the positive y-direction.
(a)Evaluate directly $\int _ { C } \vec { F } \cdot d \vec { r }$ (b)Evaluate directly $\int _ { S } \operatorname { curl }  { \vec { F } } \cdot d \vec { A }$ (c)Do these results contradict Stokes' Theorem?

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The Answer of Let $\vec { F } = 4 z \vec { i } - 4 x \vec { k }$ Let C be the circle of radius a parameterized by x = a cos t, y = 0, z = a sin t, 0 $\le$ t $\le$ 2 $\pi$ and let S be the disk in the xz-plane enclosed by C, oriented in the positive y-direction.
(a)Evaluate directly $\int _ { C } \vec { F } \cdot d \vec { r }$ (b)Evaluate directly $\int _ { S } \operatorname { curl }  { \vec { F } } \cdot d \vec { A }$ (c)Do these results contradict Stokes' Theorem?

Let F⃗=4zi⃗−4xk⃗\vec { F } = 4 z \vec { i } - 4 x \vec { k }F=4zi−4xk

Let $\vec { F } = 4 z \vec { i } - 4 x \vec { k }$