Evaluate the Surface Integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$

Short Answer

Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ for the vector field $\mathbf { F } ( x , y , z ) = x ^ { 2 } y \mathbf { i } - 3 x y ^ { 2 } \mathbf { j } + 4 y ^ { 3 } \mathbf { k }$ where S is the part of the elliptic paraboloid $z = x ^ { 2 } + y ^ { 2 } - 9$ that lies below the square $0 \leq x \leq 1 , \quad 0 \leq y \leq 1$ and has downward orientation.