Let

\vec { a } = \alpha _ { 1 } \vec { i } + \alpha _ { 2 } \vec { j } + \alpha _ { 3 } \vec { k }

be a nonzero constant vector and let

\vec { r } = x \vec { i } + y \vec { j } + z \vec { k }

.Suppose S is the sphere of radius one centered at the origin.There are two (related)reasons why

\int _ { S } ( \vec { a } \times \vec { r } ) \cdot d \vec { A } = 0

.Select them both.
A)

\int _ { S } ( \vec { a } \times \vec { r } ) \cdot d \vec { A } = 0

because

\vec { a } \times \vec { r } = \overrightarrow { 0 }

.
B)

\int _ { S } ( \vec { a } \times \vec { r } ) \cdot d \vec { A } = 0

because

\operatorname { div } ( \vec { a } \times \vec { r } ) = 0

.
C)

\int _ { S } ( \vec { a } \times \vec { r } ) \cdot d \vec { A } = 0

because

\vec { r }

is parallel to the

d \vec { A }

element everywhere on S and so

\vec { a } \times \vec { r }

is perpendicular to

d \vec { A }

on S.
D)

\int _ { S } ( \vec { a } \times \vec { r } ) \cdot d \vec { A } = 0

because

\vec { a } \times \vec { r }

is a constant vector field.
E)

\int _ { S } ( \vec { a } \times \vec { r } ) \cdot d \vec { A } = 0

because

\int _ { H } ( \vec { a } \times \vec { r } ) \cdot d \vec { A } = Q > 0

and

\int _ { L } ( \vec { a } \times \vec { r } ) \cdot d \vec { A } = - Q

, where H is the upper unit hemisphere and L is the lower unit hemisphere.

Let F⃗=5xi⃗+zj⃗+(y+5x)k⃗\vec { F } = 5 x \vec { i } + z \vec { j } + ( y + 5 x ) \vec { k }F=5xi+zj​+(y+5x)k

Essay