Suppose that $\vec { F }$ is a smooth vector field, defined everywhere.
It is possible that $\int _ { S } \vec { F } \cdot d \vec { A } = 3 r ^ { 2 } + 2 r$ , where S is a sphere of radius r centered at the origin.

Question

Quizplus · Accepted Answer

The integral of a vector field over a surface, like the flux through a sphere, should depend on the properties of the field and the geometry of the surface. However, for a sphere centered at the origin, the result of such an integral would depend on the sphere's radius in a way that reflects the symmetry and properties of the field, but an expression like $3r^2 + 2r$ suggests a specific relationship with the radius that doesn't account for the vector field's behavior or how it interacts with the sphere's surface in a physically meaningful way. The integral's dependence on the radius should reflect the geometric or physical properties of the situation, which typically would not result in such a direct polynomial relationship without further context about $\vec{F}$.

Suppose That F⃗\vec { F }F Is a Smooth Vector Field, Defined Everywhere

Suppose That $\vec { F }$ Is a Smooth Vector Field, Defined Everywhere