## Calculus: Early Transcendentals 8th Edition

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, $\iint_S F \cdot dS=\iint_D(-a(\frac{\partial g}{\partial x})-b(\frac{\partial g}{\partial x})+c)dA$