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 $\mathbf { F } ( x , y , z ) = y ^ { 2 } z \mathbf { i } + x z \mathbf { j } + x ^ { 2 } y ^ { 2 } \mathbf { k }$ and S is the part of the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ that lies inside the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ , oriented upward.