Let $\vec { G }$ be a smooth vector field with $\operatorname { div } \vec { G } = 3$ at every point in space and let S₁ and S₂ be spheres of radius r, oriented outward, centered at (0,0,0)and at (1,2,1), respectively.
$\int _ { s _ { 1 } } \vec { G } \cdot d \vec { A } = \int _ { S _ { 2 } } \vec { G } \cdot d \vec { A }$ .

Question

Let

\vec { G }

be a smooth vector field with

\operatorname { div } \vec { G } = 3

at every point in space and let S₁ and S₂ be spheres of radius r, oriented outward, centered at (0,0,0)and at (1,2,1), respectively.

\int _ { s _ { 1 } } \vec { G } \cdot d \vec { A } = \int _ { S _ { 2 } } \vec { G } \cdot d \vec { A }

.

Quizplus · Accepted Answer

The divergence theorem states that the flux of a vector field through a closed surface is equal to the integral of the divergence of the field over the volume enclosed by the surface. Since the divergence of $\vec{G}$ is constant (3) throughout space, the flux through any closed surface only depends on the volume of the space enclosed, not its location or shape. Both spheres enclose the same volume for a given radius $r$, hence the integrals are equal.

Let G⃗\vec { G }G Be a Smooth Vector Field With div⁡G⃗=3\operatorname { div } \vec { G } = 3divG=3

Let $\vec { G }$ Be a Smooth Vector Field With $\operatorname { div } \vec { G } = 3$