نظرية التباعد في الفضاء ثلاثي الأبعاد

In vector calculus, the total flow of a vector field through a surface is called flux. For closed three-dimensional shapes, calculating this at every boundary point is often a geometric challenge.

Given a vector field and a closed surface, the divergence theorem equates the total flux of the vector field through the closed surface to the volume integral of the field’s divergence within the enclosed region.

This theorem shows its utility when applied to the electric field of a central point charge extending through the hollow region bounded between an inner sphere and an irregular outer boundary.

The total surface integral of this hollow region is equal to the difference between the outer boundary's flux and the inner boundary's flux due to their opposing normal vectors.

Because no charge exists within this hollow region, the divergence of the electric field within this specific volume is exactly zero. So the outward flux through the irregular outer boundary is exactly equal to the flux through the inner sphere.

This proves that net flux depends entirely on the enclosed charge, completely independent of the outer surface's shape.