Problem. 5: Verify Stokes's Theorem for the vector field
F(x, y, z) = (-y + z) i + (x - z) j + (x - y) k
and the surface S: z = 9 - x$^2$ - y$^2$, z ? 0 with upward orientation.
• Line integral:
(1) Parameterize the boundary curve C of the surface.
r(t) =
(2) Calculate the line integral.
$\int_C$ F \cdot dr = $\int_0^{2\pi}$ ? dt
= ?