Answer
(a) See the explanation below.
(b) See the explanation below.
(c) See the explanation below.
(d) See the explanation below.
Work Step by Step
a) A surface is defined to be oriented if it consists of two sides and two normal vectors at every point over a surface $S$ which are defined as $n$ and $-n$.
The best example for non-orientable surface is: Mobius Strip, which consists of only one side.
b) The flux through an oriented surface $S$ with unit normal vector $n$ can be expressed as:
$\iint_S F \cdot dS=\iint_S F \cdot n dS$
(c) From part (b), we have
$\iint_S F \cdot dS=\iint_S F \cdot n dS$
where $dS= |r_m \times r_n|$dA
Thus, $\iint_S F \cdot dS=\iint_D F \cdot (r_m \times r_n) dA$
(d) The surface integral $f(x,y,z)$ over $S( z=g(x,y))$ can be expressed as:
Suppose $F=ai+bj+ck$
Thus, $\iint_S F \cdot dS=\iint_D(-a(\frac{\partial g}{\partial x})-b(\frac{\partial g}{\partial x})+c)dA$
