(a)Compute the flux of the vector field $\vec { F } = y \vec { i } - x \vec { j } + 5 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) z \vec { k }$ through S_a, the sphere of radius a, $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = a ^ { 2 }$ , oriented outward.
(b)Find $\lim _ { a \rightarrow 0 } \int _ { S _ { 0 } } \vec { F } \cdot d \vec { A }$ .

Question

(a)Compute the flux of the vector field

\vec { F } = y \vec { i } - x \vec { j } + 5 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) z \vec { k }

through S_a, the sphere of radius a,

x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = a ^ { 2 }

, oriented outward.
(b)Find

\lim _ { a \rightarrow 0 } \int _ { S _ { 0 } } \vec { F } \cdot d \vec { A }

.

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The Answer of (a)Compute the flux of the vector field $\vec { F } = y \vec { i } - x \vec { j } + 5 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) z \vec { k }$ through S_a, the sphere of radius a, $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = a ^ { 2 }$ , oriented outward. (b)Find $\lim _ { a \rightarrow 0 } \int _ { S _ { 0 } } \vec { F } \cdot d \vec { A }$ .

(A)Compute the Flux of the Vector Field F⃗=yi⃗−xj⃗+5(x2+y2+z2)zk⃗\vec { F } = y \vec { i } - x \vec { j } + 5 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) z \vec { k }F=yi−xj​+5(x2+y2+z2)zk

(A)Compute the Flux of the Vector Field $\vec { F } = y \vec { i } - x \vec { j } + 5 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) z \vec { k }$