$$ \nabla \cdot \vec{A} = \left[\hat{x} \frac{\partial}{\partial x} + \hat{y} \frac{\partial}{\partial y} + \hat{z} \frac{\partial}{\partial z} \right] \cdot\vec{A}$$
This is how I usually write divergence, but I have a doubt on how to evaluate expressions of the form:
$$ \hat{x} \frac{\partial }{\partial x} \cdot \vec{A}$$
It is not directly obvious that we push 'push' the $ \frac{\partial}{\partial x}$ onto $\vec{A}$ then dot product
$$ \hat{x} \cdot \frac{\partial }{\partial x} \vec{A}$$
To be explicit about my question, I find it a bit difficult that though the dot product comes , we first have to take the derivative and then dot, is there a better notation for divergence? I know the of bracket notation but that requires both the vector and divergence operator to be expressed in same basis eg: cartesian like for example if $A=<A_x,A_y,A_z>$ then:
$$ \nabla \cdot A = < \frac{\partial}{\partial x}, \frac{\partial}{\partial y} , \frac{\partial }{\partial z}> \cdot <A_x,A_y,A_z>$$
The order is clear and there is no doubt but does there exist a notation which makes it clear how to do opertions when gradient is expressed in one coordinate system and the function in another?