Appendix 6 — Derivation of Gauss's Theorem
We begin with a cube of infinitesimally small dimensions.
A flux \(\vec{F}\) flows through this infinitesimal cube.
This flux is not the same everywhere and is therefore a function of \(x,y,z,t\) .
The flux is a vector, since it has both magnitude and direction:
\begin{equation}
\vec{F}_{\text{flux}} = \vec{F}(x,y,z,t)
\label{eq:R01}
\end{equation}
Flux through a surface
Now consider the right face of the cube, a plane parallel to the \(y\)-\(z\) plane.
The flux through this surface is determined by the component of
\(\vec{F}\) that is perpendicular to that plane.
If \(\xi\) is the angle between \(\vec{F}\) and the surface, then:
\begin{equation}
\vec{F}_{\text{right}} = \vec{F} \,\sin\xi \, dy\,dz
\label{eq:R02}
\end{equation}
We represent the surface as a vector \(d\vec{A}\) , which is perpendicular to the plane:
\begin{equation}
d\vec{A} = \vec{dy} \times \vec{dz},
\qquad
|dA| = \sin\xi \, dy\,dz
\label{eq:R03}
\end{equation}
The flux through the right face then becomes:
\begin{equation}
\vec{F}_{\text{right}}
= \vec{F} \sin\xi \, dy\,dz
= \vec{F} \cos\!\left(\tfrac{\pi}{2}-\xi\right) dA
= \vec{F} \cos\varphi \, dA
= \vec{F}\cdot d\vec{A}
\label{eq:R04}
\end{equation}
Here \(d\vec{A}\) is perpendicular to the surface and
\(\varphi\) is the complementary angle of \(\xi\) .
We therefore recognize the dot product:
\begin{equation}
Flux_{right}=\vec{F}d\vec{A}\,\cos\phi=\vec{F}\cdot d\vec{A}
\label{eq:R05}
\end{equation}
Flux through the total surface of the cube
For a finite cube, the total flux is the sum of the contributions from all six faces:
\begin{equation}
\begin{aligned}
& F_{\text{flux, cube}} =\iint_{\text{right}} \vec{F} \cdot d\vec{A}+
+\iint_{\text{right}} \vec{F} \cdot d\vec{A}
+\iint_{\text{left}} \vec{F} \cdot d\vec{A}
\\ &\quad
+\iint_{\text{front}} \vec{F} \cdot d\vec{A}
+\iint_{\text{back}} \vec{F} \cdot d\vec{A}
+\iint_{\text{bottom}} \vec{F} \cdot d\vec{A}
+\iint_{\text{top}} \vec{F} \cdot d\vec{A}
\label{eq:R06}
\end{aligned}
\end{equation}
Or:
\begin{equation}
F_{\text{cube}}
= \sum_{\text{all faces}} \vec{F}\cdot d\vec{A}
\label{eq:R07}
\end{equation}
We write this as a single integral over the closed surface:
\begin{equation}
F_{\text{cube}} = \oiint_{\partial A} \vec{F}\cdot d\vec{A}
\label{eq:R08}
\end{equation}
Alternative approach: flux as a limit
In the \(x\)-direction, the incoming flux is:
\begin{equation}
F_{\text{left}} = F_x \, dy\,dz
\label{eq:R09}
\end{equation}
The flux leaving the right side is:
\begin{equation}
F_{\text{right}} = (F_x + dF_x)\, dy\,dz
\label{eq:R10}
\end{equation}
The net flux in the \(x\)-direction:
\begin{equation}
F_x^{\text{net}}
= F_{\text{right}} - F_{\text{left}}
=(F_x + dF_x)\, dy\, dz - F_x\, dy\, dz
= dF_x\, dy\,dz
\label{eq:R11}
\end{equation}
Analogously:
\begin{equation}
F_y^{\text{net}} = dF_y\, dx\,dz,
\qquad
F_z^{\text{net}} = dF_z\, dx\,dy
\label{eq:R12}
\end{equation}
The total flux through the cube:
\begin{equation}
F_{\text{cube}}
= dF_x\,dy\,dz + dF_y\,dx\,dz + dF_z\,dx\,dy
\label{eq:R13}
\end{equation}
Rewritten using partial derivatives:
\begin{equation}
F_{\text{cube}}
= \left(
\frac{\partial F_x}{\partial x}
+ \frac{\partial F_y}{\partial y}
+ \frac{\partial F_z}{\partial z}
\right) dx\,dy\,dz
\end{equation}
\begin{equation}
F_{\text{cube}}
= (\vec{\nabla}\cdot\vec{F})\, dV
\label{eq:R15}
\end{equation}
The operator ∇
\begin{equation}
\vec{\nabla}
= \frac{\partial}{\partial x}\,\hat{e}_x
+ \frac{\partial}{\partial y}\,\hat{e}_y
+ \frac{\partial}{\partial z}\,\hat{e}_z
= \left(
\frac{\partial}{\partial x},
\frac{\partial}{\partial y},
\frac{\partial}{\partial z}
\right)
\label{eq:R16}
\end{equation}
Thus equation (\ref{eq:R13}) becomes:
\begin{equation}
F_{\text{cube}} = (\vec{\nabla}\cdot\vec{F})\, dV
\label{eq:R17}
\end{equation}
Net flux through the cube
By integrating over the full volume of the cube, we obtain:
\begin{equation}
F_{\text{cube}}
= \iiint_{\text{cube}} (\nabla \cdot \vec{F})\, dV
\label{eq:R18}
\end{equation}
Gauss's Theorem
Equation (\ref{eq:R08}) gave the flux through the closed surface:
\begin{equation}
F_{\text{cube}}
= \oiint_{\partial A} \vec{F}\cdot d\vec{A}
\end{equation}
Equation (\ref{eq:R18}) gave the same flux as a volume term:
\begin{equation}
F_{\text{cube}}
= \iiint_V (\vec{\nabla}\cdot\vec{F})\, dV
\end{equation}
Since both expressions describe the same flux, it follows that:
\begin{equation}
\oiint_{\partial A} \vec{F}\cdot d\vec{A}
=
\iiint_V (\vec{\nabla}\cdot\vec{F})\, dV
\label{eq:R21}
\end{equation}
Since the volume was arbitrary (not necessarily a cube), this holds for any closed volume:
\begin{equation}
\oiint_{\partial A} \vec{F}\cdot d\vec{A}
=
\iiint_V (\vec{\nabla}\cdot\vec{F})\, dV
\label{eq:R22}
\end{equation}
This is Gauss's Theorem (also known as the Divergence Theorem).
Special case: zero flux
If the net flux through the closed surface is zero:
\begin{equation}
\oiint_{\partial A} \vec{F}\cdot d\vec{A} = 0
\end{equation}
then Gauss's theorem implies:
\begin{equation}
\iiint_V (\vec{\nabla}\cdot\vec{F})\, dV = 0
\label{eq:R24}
\end{equation}
Since the volume is arbitrary:
\begin{equation}
\vec{\nabla}\cdot\vec{F} = 0
\label{eq:R25}
\end{equation}
Written out in components:
\begin{equation}
\frac{\partial F_x}{\partial x}
+
\frac{\partial F_y}{\partial y}
+
\frac{\partial F_z}{\partial z}
= 0
\label{eq:R26}
\end{equation}
In Einstein notation (with summation over repeated index \(\alpha\)):
\begin{equation}
\frac{\partial F_\alpha}{\partial x_\alpha} = 0
\label{eq:R27}
\end{equation}
