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

[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?

 

[jvicens]

Thank you for your interpretation of Joos' development of the divergence of a vector field. Your approach seems to be a valid way of expressing the equations in a more rigorous manner.

In general, when dealing with infinitesimal quantities such as dx, dy, and dz, it is important to keep in mind that they are not real numbers but rather represent small changes in the variables x, y, and z. Therefore, it is important to introduce the concept of limits in order to make the equations more precise.

Your introduction of εx, εy, and εz is a good way to account for any deviations from the exact values of the partial derivatives. However, it is worth noting that these deviations are typically assumed to be small and can be neglected in most practical applications.

Overall, your reasoning seems to be correct and your approach is a valid way of expressing the equations in a more rigorous manner. Thank you for sharing your thoughts on this topic.