Green's Identities

FIRST GREEN IDENTITY

This identity derives from Divergence Theorem applied to the vector field

\mathbf{F}=
abla \phi

: If φ is twice Continuously Differentiable , and ψ is once continuously differentiable, on some region ''U'', then:

:

\int_U \left( \psi 
abla^2 \phi
ight)\, dV = \oint_{\partial U} \left( \psi{\partial \phi \over \partial n}
ight)\, dS - \int_U \left( 
abla \phi \cdot 
abla \psi
ight)\, dV

SECOND GREEN IDENTITY

If φ and ψ are both twice continuously differentiable on ''U'', then:

:

\int_U \left( \psi 
abla^2 \phi - \phi 
abla^2 \psi
ight)\, dV = \oint_{\partial U} \left( \psi {\partial \phi \over \partial n} - \phi {\partial \psi \over \partial n}
ight)\, dS

THIRD GREEN IDENTITY

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