Math 212
Name: __________________________________
Extra Credit Homework
If F œ ØT ß Uß VÙ is a vector field, the divergence of F is defined by the formula
divaFb œ
`T
`U
`V


.
`B
`C
`D
If I is any solid region and W is the boundary surface of I , the Divergence Theorem states that
(( F † . S œ ((( divaFb .Z
W
I
where .S points in the outward direction. (See sections 13.5 and 13.8 of the textbook.)
1. Let I be the region B#  C#  D # Ÿ ", and let W be the unit sphere B#  C#  D # œ ", with outwardpointing normal vectors. Use the Divergence Theorem to evaluate the following integral:
(( aB i  #C j  $D kb † . S.
W
(Hint: Use geometric reasoning to evaluate the triple integral that you obtain.)
2. Let I be the rectangular box defined by the inequalities
! Ÿ B Ÿ ",
! Ÿ C Ÿ #,
and
! Ÿ D Ÿ $,
and let W be the boundary of I , with normal vectors pointing outwards. Use the Divergence
Theorem to evaluate the following integral:
#
#
(( € C ß BCDß BD ¡ † .S.
W