Differenze tra le versioni di "Teorema del rotore"

 
:<math>\begin{align}
\iint_S (\nabla\times\mathbf{F}) \, \cdot \; \mathrm dSd{\mathbf {S}} &=\iint_D \left\langle (\nabla\times\mathbf{F})(\psi(u,v)) \bigg |\frac{\partial\psi}{\partial u}(u,v)\times \frac{\partial\psi}{\partial v}(u,v)\right\rangle \mathop{\mathrm du}\mathop{\mathrm dv}\\
&= \iint_D \det \left [ (\nabla\times\mathbf{F})(\psi(u,v)) \frac{\partial\psi}{\partial u}(u,v) \frac{\partial\psi}{\partial v}(u,v) \right ] \mathop{\mathrm du}\mathop{\mathrm dv}
\end{align}</math>
sicché si ottiene:
 
:<math> \iint_S (\nabla\times\mathbf{F}) \, \cdot \; \mathrm dSd{\mathbf {S}} =\iint_{D} \left( \frac{\partial P_2}{\partial u} - \frac{\partial P_1}{\partial v} \right) \mathop{\mathrm du}\mathop{\mathrm dv} </math>
 
Considerando il [[teorema di Green]], dai risultati mostrati segue la tesi.