Question

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.

          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.

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(?, ?)) ·(r?(?, ?) ×r?(?, ?)).
F(r(?, ?)) ·(r?(?, ?) ×r?(?, ?)) =
(c) Compute ∫?1^?2 F(r(?, ?)) · (r?(?, ?) × 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 div(F) dV in spherical coordinates, ∭Espherical f(?, ?, ?) d? d? d?.
∫0^∫0^∫0^ ( ) d? d? d?
Evaluate the integral.

Added by Mar T.

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