Question $5$

Use Divergence Theorem to evaluate the surface integral ${}_{S}F*dS$ where $F(x,y,z)=(xy,yz,zx)$ and $S$ with the outward pointing normal is the surface of the region bounded by the coordinate planes and the plane $x+y+z=1.$

Hints:

$(x+y)(1-x-y)+\frac{1}{2}(1-x-y{)}^2=\frac{1}{2}(1-x^2-2xy-y^2)$

$(1-x^2)(1-x)-x(1-x{)}^2-\frac{1}{3}(1-x{)}^3=\frac{2}{3}-x+\frac{1}{3}{x}^3$