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

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

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

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

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