Divergence Theorem in 3D Space

In vector calculus, flux measures the total flow of a vector field through a surface. For a closed surface in three-dimensional space, this means measuring how much of the field passes outward through every point on the boundary. Directly calculating this flux can be difficult when the surface has a complicated or irregular shape. The Divergence Theorem provides a powerful alternative by relating surface flux to behavior inside the enclosed region.

The Divergence Theorem states that the outward flux of a vector field through a closed surface is equal to the triple integral of the divergence of the field over the solid region enclosed by that surface. In symbolic form,

\begin{equation*}\iiint_V (\nabla \cdot \mathbf{F})\, dV=\iint_S \mathbf{F} \cdot \mathbf{n}\, dS\end{equation*}

Here, F is the vector field, S is the closed surface, n is the outward unit normal vector, and E is the enclosed solid region. The divergence measures the extent to which the vector field behaves like a source or sink at each point.

The theorem is especially useful for electric fields produced by point charges. Consider the hollow region between a small inner sphere surrounding a central charge and a complex outer boundary. Within this hollow region, there are no charges. Therefore, the divergence of the electric field is zero everywhere in that region.

Applying the Divergence Theorem shows that the total outward flux through the boundary of the hollow region must also be zero. This means the flux through the irregular outer surface is exactly balanced by the flux through the inner sphere, with orientation taken into account. As a result, the total electric flux through the outer boundary is independent of its shape. Whether the surface is spherical or irregular, the flux depends only on the net charge enclosed, not on the geometry of the boundary.