Use Stokes' Theorem to Evaluate $\iint _ { S } \operatorname { curl } \mathbf { F } \cdot d \mathbf { S }$

Short Answer

Use Stokes' Theorem to evaluate $\iint _ { S } \operatorname { curl } \mathbf { F } \cdot d \mathbf { S }$ where $\mathbf { F } ( x , y , z ) = \left( x + \tan ^ { - 1 } y z \right) \mathbf { i } + y ^ { 2 } z \mathbf { j } + z \mathbf { k }$ and S is the part of the hemisphere $z = \sqrt { 9 - y ^ { 2 } - z ^ { 2 } }$ that lies inside the cylinder $y ^ { 2 } + z ^ { 2 } = 4$ , oriented in the direction of the positive x-axis.