## Proof That the Flow Across Any Closed Curve is Zero in a Region of Zero Divergence

Theorem Suppose a flow
$\mathbf{F}$
is defined in a region
$B$
such that the divergence of
$\mathbf{F}$
is zero everywhere in
$\mathbf{B}$
the the rate of flow across every closed path in
$C$
is zero. Proof
Let
$C$
be any closed curve in a region
$B$
.
The rate of flow across
$C$
is
$\oint_C -f_2 (x,y) dx + f_1 (x,y_ dy$
where
$\mathbf{F} f_1 (x,y) \mathbf{i} + f_2 (x,y) \mathbf{j}$

From the theorem
$div \: \mathbf{F} = 0= \frac{\partial f_1}{\partial x} + \frac{\partial f_2}{\partial y}$

Apply Green's Theorem.
$\oint_C -f_2 (x,y) dx + f_1 (x,y_ dy = \int \int_B \frac{\partial f_1}{\partial x} + \frac{\partial f_2}{\partial y} \: dx \: dy =0$