Let $\vec { F }$ be a vector field such that $\operatorname { div } { \vec { F } } = 4 - 2 x ^ { 2 } - 2 y ^ { 2 } - 2 z ^ { 2 }$ (a)Is $\vec { F }$ a curl field?
(b)Use spherical coordinates to evaluate $\int _ { W } \operatorname { div } \vec { F } d V$ where W is the solid ball of radius R centered at the origin.
(c)Use the result of part (b)to find the radius of a sphere centered at the origin, such that the flux of $\vec { F }$ out of this sphere is zero.

Question

Let $\vec { F }$ be a vector field such that $\operatorname { div }   { \vec { F } } = 4 - 2 x ^ { 2 } - 2 y ^ { 2 } - 2 z ^ { 2 }$ (a)Is $\vec { F }$ a curl field?
(b)Use spherical coordinates to evaluate $\int _ { W } \operatorname { div } \vec { F } d V$ where W is the solid ball of radius R centered at the origin.
(c)Use the result of part (b)to find the radius of a sphere centered at the origin, such that the flux of $\vec { F }$ out of this sphere is zero.

Quizplus · Accepted Answer

The Answer of Let $\vec { F }$ be a vector field such that $\operatorname { div }   { \vec { F } } = 4 - 2 x ^ { 2 } - 2 y ^ { 2 } - 2 z ^ { 2 }$ (a)Is $\vec { F }$ a curl field?
(b)Use spherical coordinates to evaluate $\int _ { W } \operatorname { div } \vec { F } d V$ where W is the solid ball of radius R centered at the origin.
(c)Use the result of part (b)to find the radius of a sphere centered at the origin, such that the flux of $\vec { F }$ out of this sphere is zero.

Let F⃗\vec { F }F Be a Vector Field Such That div⁡F⃗=4−2x2−2y2−2z2\operatorname { div } { \vec { F } } = 4 - 2 x ^ { 2 } - 2 y ^ { 2 } - 2 z ^ { 2 }divF=4−2x2−2y2−2z2

Let $\vec { F }$ Be a Vector Field Such That $\operatorname { div } { \vec { F } } = 4 - 2 x ^ { 2 } - 2 y ^ { 2 } - 2 z ^ { 2 }$