Chapter 16.8, Problem 3PT

Let F(x, y, z) = zi + xj − yk and the curve C be the boundary of that portion of the surface S above the rectangular region D: 0 ≤ x ≤ 1, 0 ≤ y ≤ 2. Use Stokes’ Theorem to convert ${\displaystyle \underset{S}{\iint }F\cdot }dS$ to a line integral and then rewrite the line integral as a double integral.

1. a) ${\displaystyle {\int }_0^2{\displaystyle {\int }_0^1\left(2u+1\right)dvdu}}$
2. b) ${\displaystyle {\int }_0^2{\displaystyle {\int }_0^1\left(1-2u\right)dvdu}}$
3. c) ${\displaystyle {\int }_0^1{\displaystyle {\int }_0^2\left(2u+1\right)dvdu}}$
4. d) ${\displaystyle {\int }_0^2{\displaystyle {\int }_0^1\left(1-2u\right)dvdu}}$