Find the value of divergence theorem for the field D = 2xy i + x2 j for the rectangular parallelepiped given by x = 0 and 1, y = 0 and 2, z = 0 and 3.

To find the value of the divergence theorem for the given vector field ( mathbf{D} = 2xy mathbf{i} + x^2 mathbf{j} ), over the rectangular parallelepiped bounded by (x = 0) and (1), (y = 0) and (2), (z = 0) and (3), we first need to understand and apply the divergence theorem itself. The divergence theorem relates the flow (flux) of a vector field through a closed surface to the divergence of the field in the volume enclosed by the surface. Mathematically, it is represented as:

[ intintint_V (nabla cdot mathbf{D}) dV = intint_S mathbf{D} cdot mathbf{n} dS ]

Where:

– (V) is the volume inside the surface (S),

– (mathbf{D}) is the vector field,

– (nabla cdot mathbf{D}) represents the divergence of (mathbf{D}),

– (dV) is a volume element, and

– (dS) is an element of the surface area with (mathbf{n}) being the outward facing normal.

Given the vector field (mathbf{D} = 2xy mathbf{i} + x^2 mathbf