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

Multiple Choice

Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , where $\mathbf { F } = x \mathbf { i } + y \mathbf { j } + x \mathbf { k }$ and S is the part of the plane $z = 1 - x - y$ in the first octant with downward orientation.

A) $\frac { 1 } { 3 }$
B) $\frac { 1 } { 2 }$
C) $\frac { 1 } { 4 }$
D) $1$
E) $- \frac { 1 } { 3 }$
F) $- \frac { 1 } { 2 }$
G) $- \frac { 1 } { 4 }$
H) $- 1$