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

\[\mathbf{F}\]

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. ProofLet

\[C\]

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.

\[\oint_C -f_2 (x,y) dx + f_1 (x,y_ dy = \int \int_B \frac{\partial f_1}{\partial x} + \frac{\partial f_2}{\partial y} \: dx \: dy =0 \]