section_path: In calculus/Divergence theorem
section_name: Divergence theorem
paragraph_text: Divergence theorem
  
  Suppose V is a subset of R n {\displaystyle \mathbb {R} ^{n}} (in the case of n = 3, V represents a volume in 3D space) which is compact and has a piecewise smooth boundary S (also indicated with ∂ V = S). If F is a continuously differentiable vector field defined on a neighborhood of V, then the divergence theorem says:
  
  The left side is a volume integral over the volume V, the right side is the surface integral over the boundary of the volume V. The closed manifold ∂ V is quite generally the boundary of V oriented by outward-pointing normals, and n is the outward pointing unit normal field of the boundary ∂ V. (d S may be used as a shorthand for n dS.)