Which are the most used and correct nomenclature for gradient, divergence, curl and Laplace operator in differents contexts? I used to write these operator in this way:
$\vec{\nabla}$  for divergence and gradient and for Laplace operator $\vec{\nabla}^2$.
But I have noticed that in some books and website 
divergence is written also like : $\mathbf{div}$  or with $\nabla$ non signed as a vector  (it wasn't bold) and Laplace operator as $\Delta$.
So my question is: there is a preferred nomenclature in some contexts or it is indifferent?
 A: Nabla symbol $\nabla$ by itself is used for the gradient only. The use of the arrow is more of a physicist's style and is optional. It is only a question of personal taste.
You can also use $\mathbf{grad}$ instead of $\nabla$. In the origin they used $\mathbf{slope}$ instead of $\mathbf{grad}$ but now it is not used any more.
For divergence and curl, you can use simply $\mathbf{div}$ and $\mathbf{curl}$ respectively or you can use the nabla symbol followed by the dot product and the cross product respectively. In the origin there was also the convergence whose symbol was $\mathbf{conv}$ and it was the negative of the divergence. (See Oliver Heaviside, the engineer who invented them, among lots of indispensable things). It is not used anymore.
The curl was and is also known as rotor or rotational and an alternative symbol is $\mathbf{rot}$. This symbol is not often used at this time in english-speaking countries, but it is in widespread use elsewhere.
Also for the Lapace operator, both of the symbols you said are used (with or without the arrow) depending on personal taste.
Of course one has to be consistent.
