As I began to teach myself in differential geometry, I finally used to use the Nabla-Operator.
I know and understand its usage as in
$$ \nabla f := \left( \begin{matrix} \frac{∂f}{∂x_1} & \frac{∂f}{∂x_2} & \cdots & \frac{∂f}{∂x_n} \end{matrix} \right)^\intercal $$
but in many books I read a pure definition of $\nabla$: $$ \nabla := \left( \begin{matrix} \frac{∂}{∂x_1} & \frac{∂}{∂x_2} & \cdots & \frac{∂}{∂x_n} \end{matrix} \right) ^ \intercal $$
which seems to be just a visualisation of the content, because it's mathematically false $-$ an equation needs to have two evaluatable terms on both sides, but an operator is not a value.
For example, the derivation operator can conformly be defined as $$ \frac{∂}{∂x_i}: ℝ → ℝ, \quad f ↦ \frac{∂f}{∂x_i} := \lim_{x_i→0}{\frac{f(x_1,\cdots,x_i+h,\cdots,x_n)-f(_1,\cdots,x_i,\cdots,x_n)}{h}} $$
But the Nabla-Operator is applied in multiple ways; therefore, one cannot define it as a function.
Do I suppose rightly that there does not exists an explicit definition, or does there exist some kind of ‘trick’?
The keyword is Operator Calculus or alternatively Operational Calculus. Here is an introductory (PDF) document . Other references are easily found on the internet, such as Fractional Calculus (Wikipedia), What is operator calculus? (MSE), How to make sense of this calculus notation, Advanced College Level (MSE).