Suppose That S Is the Surface Which Is a Portion $z = f ( x , y )$

Essay

Suppose that S is the surface which is a portion of the graph of a smooth function $z = f ( x , y )$ over a region R in the xy-plane, oriented upward.Consider the vector field $\vec { F } = f _ { x } ( x , y ) \vec { i } + f _ { y } ( x , y ) \vec { j } + g ( x , y , z ) \vec { k }$ .
Find $g ( x , y , z )$ so that $\int _ { S } { \vec { F } } \cdot d \vec { A } = 0$ .