Divergence theorem

The divergence theorem states that the integral of the Divergence of a three-dimensional Vector field in a volume is equal to the integral of the field itself projected over the bounding Surface of the volume. For a generic vector field F\mathbf{F} in a volume VV of bounding surface SS, the theorem states:

∫V∇⋅F=∮SF⋅da\int_{V}\nabla\cdot\mathbf{F}=\oint_{S}\mathbf{F}\cdot d\mathbf{a}

where da=n^ dad\mathbf{a}=\hat{\mathbf{n}}\ da is the area element projected with the normal unit vector n^\hat{\mathbf{n}}.

Mathematical treatment

Let F:U⊂R3→R3F : U ⊂ \mathbb{R}^{3} → \mathbb{R}^{3} be a vector field. Let's consider a closed domain D⊂UD ⊂ U of boundary ∂+D=Σ∂^+D = \Sigma, which is the support of a closed and oriented surface with an outward normal field ν^\hat{\nu}. Then

∭D(∇⋅F)(x,y,z) dxdydz=∬∂+DF⋅ν^  dS\iiint_{D}(\nabla\cdot F)(x,y,z)\ dxdydz=\iint_{\partial^+D}F\cdot\hat{\nu}\;dS

A similar, more general result holds in other dimensions too, where the integral goes from an NN-dimensional integral to an (N−1)(N-1)-dimensional integral. For instance, in the plane it goes from a surface to the bounding Curve. In one-dimension, it becomes integration by parts.

Despite the name, similar results also hold for the Curl

∭D(∇×F)(x,y,z)dxdydz=−∬∂+DF×ν^  dS\iiint_{D}(\nabla\times F)(x,y,z)dxdydz=-\iint_{\partial^+D}F\times\hat{\nu}\;dS

and the Gradient of a Scalar field ϕ:U⊂R3→R\phi:U\subset \mathbb{R}^{3}\rightarrow \mathbb{R}

∭D(∇ϕ)(x,y,z)dxdydz=∬∂+Dϕν^  dS\iiint_{D}(\nabla\phi)(x,y,z)dxdydz=\iint_{\partial^+D}\phi\hat{\nu}\;dS