I am learning vector calculus form Advanced Engineering Mathematics (By Erwin Kreyzeg) and I often see statements like this :-
(1) is $\operatorname{div}(V)$, where $V = [V_1, V_2, V_3]$
Invariance of the Divergence -
The divergence $\operatorname{div}(V)$ is a scalar function, that is, its values depend only on the points in space (and, of course, on $V$) but not on the choice of the coordinates in (1), so that with respect to other Cartesian coordinates $(x^*, y^*, z^*)$ and corresponding components $V^* = [V_1^*, V_2^*, V_3^*]$ of $V$, then $\operatorname{div}(V) = \nabla^* \cdot V^*$.
What i would like to know it that what does the author means by independent of choice of co-ordinates? can someone explain it with an example?
Similar things were said for gradient too, but that time i just ignored it cuz it was difficult to comprehend.
What the author means is that if you have two systems of coordinates $(x_1,x_2,x_3)$ and $(x'_1,x'_2,x'_3)$ respectively for the same domain of space, then the divergence of the field $V$ as represented in space $x_1x_2x_3$ is not different from the divergence of the same field, but represented as $V^*=V'$ in space $x'_1x'_2x'_3.$
The claim, written out explicitly, is that $$\sum_{i=1}^3 \frac{\partial V_i}{\partial x_i}=\sum_{j=1}^3 \frac{\partial V'_j}{\partial x'_j}.$$
Although this is beside the point here, an intuitive reason for this is that a scalar field over a region of physical space does not depend on how we look at (coordinatize) the space. So obviously its divergence also should be objective -- independent of our frame of reference or point of view.