Let E be the solid ball of radius 4 centered on the origin of R³ and let F = (x + z)i - (y - z)j + (y - x)k.
(a) Parameterize S, the boundary of E, using spherical coordinates r(?, ?) so that the induced orientation is
outward.
r(?, ?) =
(b) Compute F(r(?, ?)) \cdot (r_?(?, ?) \times r_?(?, ?)).
F(r(?, ?)) \cdot (r_?(?, ?) \times r_?(?, ?)) =
(c) Compute $\int_{?_1}^{?_2} F(r(?, ?)) \cdot (r_?(?, ?) \times r_?(?, ?)) \, d?$, where $?_1$ is the lower bound of ? and $?_2$ is the upper
bound of ?.
(d) Compute the flux of F across S.
(e) Compute div(F).
div(F) =
(f) Write the integral $\iiint_E$ div(F) dV in spherical coordinates, $\iiint_{E_{spherical}} f(?, ?, ?) \, d? \, d? \, d?$.
$\int_0^{ } \int_0^{ } \int_0^{ } ( ) \, d? \, d? \, d?$
Evaluate the integral.
