Chapter 17.6, Problem 49E

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a.    If the surface S is given by $\left\{\left(x,y,z\right):0\le x\le 1,0\le y\le 1,z=10\right\}$, then ${\displaystyle {\iint }_{S}f\left(x,y,z\right)}dS={\displaystyle {\int }_0^1{\displaystyle {\int }_0^{1}f\left(x,y,10\right)dxdy}}$.

b.    If the surface S is given by $\left\{\left(x,y,z\right):0\le x\le 1,0\le y\le 1,z=x\right\}$, then ${\displaystyle {\iint }_{S}f\left(x,y,z\right)dS=}{\displaystyle {\int }_0^1{\displaystyle {\int }_0^{1}f\left(x,y,z\right)}}dxdy$.

c.    The surface r = (v cos u, v sin u, v²), for $0\le u\le \pi ,0\le v\le 2$, is the same as the surface $r=\langle \sqrt{v}\cos 2u,\sqrt{v}\sin 2u,v\rangle$, for $0\le u\le \pi /2,0\le v\le 4$.

d.    Given the standard parameterization of a sphere, the normal vectors t_u × t_v are outward normal vectors.