What is the name for this definition of nabla using integral? I was taught that operator $\nabla$ can be defined as follows:
$$
\nabla = \lim\limits_{\delta\rightarrow 0} \frac1{\mu V} \oint\limits_{\partial V} n \, \mathrm{d}S
$$
where $V$ is a region of space, $\partial V$ is it's boundary, $\mu V$ is the volume, $\delta$ is the diameter and $n$ is the normal vector on the boundary. And the operator works by taking whatever comes after it and inserting it after $n$.
Unfourtunately, I can find nothing on this formula (not even the formula itself) in English. We called this the invariant definition in Latvian. Spanish and Italian wikipedia seem to call it the intrinsic definition. Dutch seem to call it coordinate agnostic definition. But googling these terms combined with "nabla" don't produce any relevant results in English.
 A: It's perhaps more natural, due to divergence/Gauss theorem, to define divergence (and not del, at least at first) similar to what you present. If you go to Divergence wikipedia page, you'll find this definition (which is mentioned every once ina while, specially in physics books on electromagnetism). This may be referred to as the "coordinate-free" definition of divergence:
$$
\nabla\cdot \mathbf{F}:=\mathrm{div} (\mathbf{F}):=
\lim_{\delta\to 0}\frac{1}{\mu V}\int_{\partial V} \mathbf{n}\cdot \mathbf{F}dS
$$
Furthermore, one can define $\mathrm{grad}(f):=\nabla f$ and $\mathrm{curl}(\mathbf{F}):=\nabla\times \mathbf{F}$, as the unique vectors such that for all constant vectors $\mathbf{c}$, we have
$$
\mathrm{div}(f\mathbf{c})=\mathrm{grad}(f)\cdot \mathbf{c}, \qquad \mathrm{div}(\mathbf{F}\times \mathbf{c}):=\mathrm{curl}(\mathbf{F})\cdot \mathbf{c}
$$
One can then use the above coordinate-free definitions for divergence, gradient and curl, to give a corodinate-free definition for $\nabla$ which is exactly as you defined.
