Answer
See proof
Work Step by Step
Step 1
(1) $\vec{F}$ is a three-dimensional vector field.
(2) $\vec{F}$ has continuous partial derivatives.
(3) $\vec{F}$ has a divergence of $0$ at all points.
The last piece of information is the key here.
Step 2
In order to identify the divergence at a point, we draw a small sphere around it and treat the vector field as the flow of water:
(1) If the net flow of the water is positive inside that sphere, then the divergence is positive.
(2) If the net flow of the water is negative inside that sphere, then the divergence is negative.
In this case, the divergence is zero everywhere. This means that all the vector fields point in the same direction.
Step 3
Since $\sigma$ is a sphere, and $\vec{F}$ points in the same direction at all points, there is a circle such that:
(1) It shares the same radius and center as the sphere.
(2) The plane containing this circle is perpendicular to the common direction of all the vector fields.
$\vec{F}$ is tangent to the sphere at all points on this circle. Since a circle has infinitely many points, the requested proof has been delivered.
