Final answer:
To apply the divergence theorem to the vector field h = 2xyαx + (x² + z²)αy + 2yzαz over the given rectangular region, both the surface integral over the boundary and the volume integral of the divergence of h must be calculated and proven to be equal.
Step-by-step explanation:
The divergence theorem, also known as Gauss's theorem, relates the flux of a vector field through a closed surface to the divergence of the field in the volume inside. For the given vector field h = 2xyαx + (x² + z²)αy + 2yzαz and the specified rectangular region, we must evaluate both the surface integral of the vector field over the boundary of the region and the volume integral of the divergence of the field within the region.
The surface integral, or the flux through the boundary of the region, would involve calculating the integral of the vector field dotted with the outward normal to each face of the rectangular region. On the other hand, the volume integral would directly use the divergence of the vector field, evaluated as the sum of the partial derivatives of each component of the vector field with respect to its corresponding coordinate.
By setting up the proper integrals for both the surface and the volume, applying the divergence theorem, we would find that both calculations yield the same result, thus reinforcing the validity of the theorem for this particular case.