# Divergence as the limit of a surface integral a volume->0

1. Dec 16, 2015

### Odious Suspect

The following is my interpretation of the development of the divergence of a vector field given by Joos:

$$dy dz dv_x=dy dz\left(v_x(dx)-v_x(0)\right)=dy dz\left(v_x(0)+dx\frac{\partial v_x}{\partial x}(0)- v_x(0)\right)$$
$$=dy dz dx\frac{\partial v_x}{\partial x}(0)=d\tau \frac{\partial v_x}{\partial x}(0)$$
$$\oint \mathfrak{v}\cdot d\mathfrak{S}=\left(\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z}\right)d\tau$$
$$\oint \mathfrak{v}\cdot d\mathfrak{S}=d\tau \overset{\rightharpoonup }{\nabla}\cdot \mathfrak{v}$$

I know I am picking nits here, but I want to understand what a rigorous development would be. I contend that $dx, dy, dz$ are independent real number variables of arbitrary magnitude, and $dv_x\equiv dx \frac{\partial v_x}{\partial x}$. Also $\Delta v_x \equiv v_x(dx)-v_x(0)$.

Introducing $\varepsilon_x \equiv \frac{\Delta v_x}{\Delta x}-\frac{\partial v_x}{\partial x}$ and $\Delta x = dx$, we can write the first equation as:

$$dy dz dv_x=dy dz\left(v_x(dx)-v_x(0)-\varepsilon_x dx\right)=dy dz\left(v_x(0)+dx\frac{\partial v_x}{\partial x}(0)- v_x(0)\right)$$

Using $\Delta \mathfrak{v} \equiv \mathfrak{v}(dx \hat{ i } + dy \hat{ j } + dz \hat{ k })-\mathfrak{v}(0)$, the surface integral should be:

$$\oint \mathfrak{v}\cdot d\mathfrak{S}=d\tau \Delta \mathfrak{v} \cdot ( \hat{ i } + \hat{ j } + \hat{ k })$$

Using the same approach as above, we could write this as:

$$\oint \mathfrak{v}\cdot d\mathfrak{S}=\left(\frac{\partial v_x}{\partial x}+\varepsilon_x+\frac{\partial v_y}{\partial y}+\varepsilon_y+\frac{\partial v_z}{\partial z}+\varepsilon_z\right)d\tau$$

In order to make the final original equation rigorous, we would need to express it as a limit.
$$\lim_{d\tau \rightarrow 0}\frac{1}{d\tau}\oint \mathfrak{v}\cdot d\mathfrak{S}= \overset{\rightharpoonup }{\nabla}\cdot \mathfrak{v}$$

This would be much easier to follow if I could provide drawings, etc. Is my reasoning correct in what I have presented here?

2. Dec 21, 2015