Final Answer:
![\[ \int\int\int_S \mathbf{F} \cdot \mathbf{N} \, dS = (\pi)/(2) \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/awp92vdu9udow45j8yscdgynuic5ro021s.png)
The divergence theorem, applied to the given vector field
over the unit sphere (S), yields a triple surface integral of zero over the volume enclosed by (S). However, on the surface (S), the flux of
is
.
Step-by-step explanation:
To evaluate the triple surface integral, we'll use the divergence theorem, which relates a triple integral over a region to a surface integral over the boundary of the region. The divergence theorem states that
, where \(V\) is the region bounded by the surface (S),
is the vector field,
is the outward unit normal vector to (S), and \(dS\) is the surface area element.
In this case, the given vector field
is divergence-free, as
. Therefore, the triple integral reduces to
. However, on the surface of the unit sphere (S), the vector field
has a non-zero flux.
The divergence theorem is a powerful tool in vector calculus that relates volume integrals to surface integrals. It is derived from the fundamental theorem of calculus for line integrals and provides a way to simplify calculations by converting a volumetric problem into a surface problem.