Green's Identities

Green's identities are a set of three vector derivative/integral identities which can be derived starting with the vector derivative identities

$$\nabla\cdot(\psi\nabla\phi) = \psi\nabla^2\phi+(\nabla\psi)\cdot(\nabla\phi)$$ (1)

and

$$\nabla\cdot(\phi\nabla\psi) = \phi\nabla^2\psi+(\nabla\phi)\cdot(\nabla\psi),$$ (2)

where $\nabla\cdot$ is the Divergence, $\nabla$ is the Gradient, $\nabla^2$ is the Laplacian, and ${\bf a}\cdot{\bf b}$ is the Dot Product. From the Divergence Theorem,

$$\int_V (\nabla\cdot{\bf F})\,dV = \int_S {\bf F}\cdot d{\bf a}.$$ (3)

Plugging (2) into (3),

$$\int_S \phi(\nabla\psi)\cdot d{\bf a} = \int_V [\phi\nabla^2\psi+(\nabla\phi)\cdot(\nabla\psi)]\,dV.$$ (4)

This is Green's first identity.

Subtracting (2) from (1),

$$\nabla\cdot(\phi\nabla\psi-\psi\nabla\phi) = \phi \nabla^2\psi -\psi \nabla^2\phi.$$ (5)

Therefore,

$$\int_V(\phi\nabla^2\psi-\psi\nabla^2\phi)\,dV=\int_S (\phi\nabla\psi-\psi\nabla\phi)\cdot d{\bf a}.$$ (6)

This is Green's second identity.

Let $u$ have continuous first Partial Derivatives and be Harmonic inside the region of integration. Then Green's third identity is

$$u(x,y)={1\over 2\pi} \oint_C \left[{\ln\left({1\over r}\righ... ...\partial\over\partial n}\ln\left({1\over r}\right)}\right]\,ds$$ (7)

(Kaplan 1991, p. 361).

References

Kaplan, W. Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, 1991.