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