How to interpret the curl and div geometrically?

Question

How to interpret the curl and div geometrically?

My book said the vector derivative operator '$\nabla $' is considered as vector as follows :

$$\left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right)$$

Let $\mathbf F$ be a vector field.

First, I know the fact that the $\operatorname{curl}$ is the tendency of rotation and the pivot is $\operatorname{curl} \mathbf F$

However, why the curl is the tendency of rotation geometrically?

Easily, when calculating the outer product between two vector, we can interpret the result. The value is $\mathbf a\mathbf b\sin\theta$ and the direction is perpendicular to the two vectors.

I want to know to draw $\nabla$ as vector.

Second, the $\operatorname{div}$ is tendency of divergence or convergence.

But which direction? If the result of $\operatorname{div} \mathbf F$ is plus, then the tendency is same direction to $\mathbf F$? (If minus, then opposite direction?)

Answer 1

There are intuitive geometrical and limit interpretations of ${\rm div}$ and ${\rm curl}$ (dealt with in many MSE questions), but you cannot interpret $\left({\partial\over\partial x},{\partial\over\partial y},{\partial\over\partial z}\right)$ (whatever that means) or $\nabla$ as a vector in ${\mathbb R}^3$. Think of it: If these typographical objects are considered to have meaning before any vector field is given, their denoting a certain vector would imply that there are $\geq1$ distinguished vectors in ${\mathbb R}^3$, predefined by the IMU, which is absurd.