## Proof That The Divergence of the Curl of a Vector is Zero

The Divergence Theorem states that for any vector field
$\mathbf{A}$
with differentiable components defined on a volume
$V$
with boundary
$S$

$\int \int \int_V \mathbf{\nabla} \cdot \mathbf{A} dV = \int \int_S \mathbf{A} \cdot \mathbf{n} dS$

Let
$\mathbf{A} = \mathbf{\nabla} \times \mathbf{F}$
for some vector field
$\mathbf{F}$

Then
$\int \int \int_V \mathbf{\nabla} \cdot (\mathbf{\nabla} \times \mathbf{F}) dV = \int \int_S (\mathbf{\nabla} \times \mathbf{F}) \cdot \mathbf{n} dS$

Then the right hand side is zero. Hence
$\int \int \int_V \mathbf{\nabla} \cdot (\mathbf{\nabla} \times \mathbf{F}) dV =0$

The surface is arbitrary and so is the volume, hence
$\mathbf{\nabla} \cdot (\mathbf{\nabla} \times \mathbf{F}) =0$