(a)Is $\vec { F } = F _ { 1 } ( y , z ) \vec { i } + F _ { 2 } ( x , z ) \vec { j } + F _ { 3 } ( x , y ) \vec { k }$ a divergence free vector field?
(b)Do all divergence free vector fields have the form of the vector field in (a)?
(c)If $\vec { F }$ has the form given in (a)can we conclude that $\int _ { S } \vec { F } \cdot \vec { d A } = 0$ for any closed surface S?

Question

(a)Is $\vec { F } = F _ { 1 } ( y , z ) \vec { i } + F _ { 2 } ( x , z ) \vec { j } + F _ { 3 } ( x , y ) \vec { k }$ a divergence free vector field?
(b)Do all divergence free vector fields have the form of the vector field in (a)?
(c)If $\vec { F }$ has the form given in (a)can we conclude that $\int _ { S } \vec { F } \cdot \vec  { d A } = 0$ for any closed surface S?

Quizplus · Accepted Answer

The Answer of (a)Is $\vec { F } = F _ { 1 } ( y , z ) \vec { i } + F _ { 2 } ( x , z ) \vec { j } + F _ { 3 } ( x , y ) \vec { k }$ a divergence free vector field?
(b)Do all divergence free vector fields have the form of the vector field in (a)?
(c)If $\vec { F }$ has the form given in (a)can we conclude that $\int _ { S } \vec { F } \cdot \vec  { d A } = 0$ for any closed surface S?

(A)Is F⃗=F1(y,z)i⃗+F2(x,z)j⃗+F3(x,y)k⃗\vec { F } = F _ { 1 } ( y , z ) \vec { i } + F _ { 2 } ( x , z ) \vec { j } + F _ { 3 } ( x , y ) \vec { k }F=F1​(y,z)i+F2​(x,z)j​+F3​(x,y)k

(A)Is $\vec { F } = F _ { 1 } ( y , z ) \vec { i } + F _ { 2 } ( x , z ) \vec { j } + F _ { 3 } ( x , y ) \vec { k }$