About the definition of divergence

Question

The divergence is defined as:

$\nabla . \mathbf{A}=\lim \limits_{V \to 0} \dfrac{ \unicode{x222F}_{\partial V} \mathbf{A}.d\mathbf{S}}{V}$

My question is of two parts:

$(1)$ If we are using Cartesian coordinate system, should $V$ be necessarily a cuboid?

$(2)$ If we are using another coordinate system, should $V$ be necessarily a cuboid?

Answer 1

If $\bf A$ is $C^1$ and the border of $V$ is piecewise $C^1$, using the divergence theorem you can prove that this definition is equivalent to the alternative definition $\sum\bf\partial_i A_i$.