Let S be the parametric surface $x = r \cos \theta , y = r \sin \theta , z = \theta , 0 \leq r \leq 1,0 \leq \theta \leq \frac { \pi } { 2 }$ . Use Stokes' Theorem to evaluate $\iint _ { S } \operatorname { curl } \mathbf { F } \cdot d \mathbf { S }$ , where $\mathbf { F } ( x , y , z ) = - y \mathbf { i } + x \mathbf { j } + z \mathbf { k }$ .

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The Answer of Let S be the parametric surface $x = r \cos \theta , y = r \sin \theta , z = \theta , 0 \leq r \leq 1,0 \leq \theta \leq \frac { \pi } { 2 }$ . Use Stokes' Theorem to evaluate $\iint _ { S } \operatorname { curl } \mathbf { F } \cdot d \mathbf { S }$ , where $\mathbf { F } ( x , y , z ) = - y \mathbf { i } + x \mathbf { j } + z \mathbf { k }$ .

Let S Be the Parametric Surface x=rcos⁡θ,y=rsin⁡θ,z=θ,0≤r≤1,0≤θ≤π2x = r \cos \theta , y = r \sin \theta , z = \theta , 0 \leq r \leq 1,0 \leq \theta \leq \frac { \pi } { 2 }x=rcosθ,y=rsinθ,z=θ,0≤r≤1,0≤θ≤2π​

Let S Be the Parametric Surface $x = r \cos \theta , y = r \sin \theta , z = \theta , 0 \leq r \leq 1,0 \leq \theta \leq \frac { \pi } { 2 }$