\[\mathbf{\nabla} o = lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_{\delta S} d \mathbf{S} o\]
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.\[\begin{equation} \begin{aligned} \mathbf{\nabla} o \phi &= lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int{\delta S} d \mathbf{S} o \phi \\ &= lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_{\delta S} \mathbf{n} \phi o dS \end{aligned} \end{equation}\]
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.\[\begin{equation} \begin{aligned} \mathbf{\nabla} o \mathbf{F} &= lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int{\delta S} d \mathbf{S} o \mathbf{F} \\ &= lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_{\delta S} \mathbf{n} \cdot \mathbf{F} dS \end{aligned} \end{equation}\]
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.\[\begin{equation} \begin{aligned} \mathbf{\nabla} o \mathbf{F} &= lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int{\delta S} d \mathbf{S} o \mathbf{F} \\ &= lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_{\delta S} \mathbf{n} \times \mathbf{F} dS \end{aligned} \end{equation}\]
which is equivalent to
\[\mathbf{\nabla} \times \mathbf{F} = lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_{\delta S} \mathbf{n} \times \mathbf{F} dS\]