## Definition of an Operator

The equation
$\mathbf{\nabla} o = lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_{\delta S} d \mathbf{S} o$
where
$\delta S$
is the surface of the region
$\delta V$
defines an operator, where
$o$
is ordinary multiplication, dot or cross product.
If
$o$
is ordinary multiplication then the operator acts on a scalar field.
\begin{aligned} \mathbf{\nabla} o \phi &= lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int{\delta S} d \mathbf{S} o \phi \\ &= lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_{\delta S} \mathbf{n} \phi o dS \end{aligned}

which is equivalent to
$\mathbf{\nabla} \phi = lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_{\delta S} \mathbf{n} \phi dS$

If
$o$
a dot product then the operator acts on a vector field.
\begin{aligned} \mathbf{\nabla} o \mathbf{F} &= lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int{\delta S} d \mathbf{S} o \mathbf{F} \\ &= lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_{\delta S} \mathbf{n} \cdot \mathbf{F} dS \end{aligned}

which is equivalent to
$\mathbf{\nabla} \cdot \mathbf{F} = lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_{\delta S} \mathbf{n} \cdot \mathbf{F} dS$

If
$o$
a cross product then the operator acts on a vector field.
\begin{aligned} \mathbf{\nabla} o \mathbf{F} &= lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int{\delta S} d \mathbf{S} o \mathbf{F} \\ &= lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_{\delta S} \mathbf{n} \times \mathbf{F} dS \end{aligned}

which is equivalent to
$\mathbf{\nabla} \times \mathbf{F} = lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_{\delta S} \mathbf{n} \times \mathbf{F} dS$