differentiation - Intuitive analysis of gradient, divergence, curl

I have read the most basic and important parts of vector calculus are gradient, divergence and curl. These three things are too important to analyse a vector field and I have gone through the physical meaning of gradient, divergence, and curl.

Gradient indicates the direction of maximum rate of change of a function, and it is defined as $\nabla u(\mathbf{x})\equiv\mathrm{grad} \, u(\mathbf{x})$ where $u(\mathbf{x})$ is a scalar function of the position vector $\mathbf{x}$.

Whereas divergence represents the volume density of the outward flux from an infinitesimal volume around a point and is defined as $\boldsymbol{\nabla} \boldsymbol{\cdot} \mathbf{A}(\mathbf{x})\equiv\mathrm{div} \, \mathbf{A}(\mathbf{x})$

And the curl signifies the infinitesimal rotation around a point and is defined as $\boldsymbol{\nabla} \boldsymbol{\times} \mathbf{A}(\mathbf{x})\equiv \mathrm{curl} \, \mathbf{A}(\mathbf{x})$

And now I have study and also mathematically formulate that $\boldsymbol{\nabla} \boldsymbol{\cdot} \left[\boldsymbol{\nabla} \boldsymbol{\times} \mathbf{A}(\mathbf{x})\right]=\mathrm{div} \left[\mathrm{curl} \mathbf{A}(\mathbf{x})\right]=0$.

And $\boldsymbol{\nabla} \boldsymbol{\times}\left[\nabla u(\mathbf{x})\right] =\mathrm{curl}\left[ \mathrm{grad} \, u(\mathbf{x})\right]= \boldsymbol{0}$ where u is a scalar function

Now my question is how can I interpret those two expressions? I mean, I want the physical significance of these two expressions in the sense of definition of gradient, divergence, curl, and also using graphical representation.

Answer

Let us begin with $$\mathrm{div} \left[\mathrm{curl} \mathbf{A}(\mathbf{x})\right]=0$$

Consider the electrostatic field, it has sources to emerge from (positive charges) and sinks to go into (negative charges). Such a field has a non-zero divergence.

When we say that the divergence of $ \mathrm{curl} \mathbf{A}(\mathbf{x})$ is equal to zero, this means that the curl doesn't have any sources or sinks, its total flux out of a closed surface is always zero and it is usually either a uniform field or forms closed vortices (as the magnetic field). In general, any vector field with a zero divergence has to be a curl of some other field.

The second equation:$$\mathrm{curl}\left[ \mathrm{grad} \, u(\mathbf{x})\right]= \boldsymbol{0}$$ doesn't have much to deduce from. A field with zero curl is called irrotational. In a simply connected region, such a field is conservative (path independence and no vortices), but we already know this for a gradient.