Green's Theorem

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Green's first and second Theorem are two usefule identities that are used over and over in multivariable problems (e.g. Poisson's Equation). They are basically the result of playing with Gauß's Theorem. Consider two scalar functions $\phi$ and $\psi$ continuous differentiable in the volume $V$ bound by the surface $S$. We start with the vector field $\phi \nabla \psi$. By applying Gauß's theorem we find \begin{align} \oint_S \phi \, \nabla \psi \, d\mathbf{S}=\int_V \nabla \cdot (\phi \,\nabla \psi)\, dV=\int_V \left[ \phi \, \nabla^2 \psi+ \nabla \phi \, \nabla \psi \right] \,dV \end{align} writing $\phi \, \nabla \psi \, d\mathbf{S}$ as $\phi \, \nabla \psi \, \hat{\mathbf{n}} \, dS$ and using $\nabla \psi \, \hat{\mathbf{n}}=\frac{\partial \psi}{\partial n}$ we arrive at the identity called

Green's first identity \begin{align} \oint_S \phi \, \frac{\partial \psi}{\partial n} \, dS=\int_V \left[ \phi \, \nabla^2 \psi+ \nabla \phi \, \nabla \psi \right] \,dV \end{align}

switching the role of $\phi$ and $\psi$ in Green's first identity gives \begin{align} \int_V \nabla \cdot (\psi \, \nabla \phi) dV = \int_V \left[ \psi \, \nabla^2 \phi+ \nabla \psi \, \nabla \phi \right] \, dV \end{align}

if we subtract both equations from each other we arrive at

Green's second identity \begin{align} \oint_S \left[ \phi \, \frac{\partial \psi}{\partial n} -\psi \, \frac{\partial \phi}{\partial n}\, \right] dS=\int_V \left[ \phi \, \nabla^2 \psi-\psi \, \nabla^2 \phi \right] \,dV \end{align}