Answer
See proof
Work Step by Step
Step 1 We know that the divergence of a vector field $\text{div}(\vec{F})$ takes the vector value $\vec{F}$ and turns it into a number $\text{div}(\vec{F})$. For example, in a Cartesian coordinate system, we have: \[ \text{div}(\vec{F}) = \frac{\partial}{\partial x}(F_x) + \frac{\partial}{\partial y}(F_y) + \frac{\partial}{\partial z}(F_z). \] Step 2 Also, we know that the flux of a vector field $\vec{F}$ over a closed surface $\Sigma$ is related to the divergence of $\vec{F}$ through the Divergence Theorem: \[ \iint_{\Sigma} \vec{F} \cdot \mathbf{n} \, dS = \iiint_{V} \text{div}(\vec{F}) \, dV, \] where $V$ is the volume enclosed by $\Sigma$. Step 3 Let us find the divergence of $\vec{F}$ at a certain point $\mathbf{P}_0$. We can take the volume $V$ in Step 2, and thus the surface $\Sigma$ as well, to be a really small volume around the point $\mathbf{P}_0$. If $\text{div}(\vec{F})$ does not change discontinuously, we can approximate $\text{div}(\vec{F})$ to be a constant in the small volume $V$, thus Step 2 gives us: \[ \iint_{\Sigma} \vec{F} \cdot \mathbf{n} \, dS = \iiint_{V} \text{div}(\vec{F}) \, dV \approx \text{div}(\vec{F}(\mathbf{P}_0)) \cdot \text{vol}(V), \] where $\text{vol}(V)$ is the volume of the region $V$. Step 4 From the approximation above, we can define the divergence of $\vec{F}$ at the point $\mathbf{P}_0$ as: \[ \text{div}(\vec{F}(\mathbf{P}_0)) \approx \frac{1}{\text{vol}(V)} \iint_{\Sigma} \vec{F} \cdot \mathbf{n} \, dS. \] The key thing in this approach is that we have approximated $\text{div}(\vec{F}(\mathbf{P}_0))$ to be constant. To make this approximation more valid, we can take the limit of $\text{vol}(V) \to 0$: \[ \text{div}(\vec{F}(\mathbf{P}_0)) = \lim_{\text{vol}(V) \to 0} \frac{1}{\text{vol}(V)} \iint_{\Sigma} \vec{F} \cdot \mathbf{n} \, dS. \] The independence on the coordinate system means that the $\text{div}(\vec{F})$ at the point $\mathbf{P}_0$ gives the same value for each coordinate system in which we express $\vec{F}$ and Step 1. The right side of the above definition of the divergence of $\vec{F}$ is called the flux density and is independent of the coordinate system. Since the definition above comes from the Divergence Theorem, which holds for the initial definition of the divergence (Step 1, for example), we say that the divergence in general is independent of the coordinate system.
