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

 del ·(psidel phi)=psidel ^2phi+(del psi)·(del phi)


 del ·(phidel psi)=phidel ^2psi+(del phi)·(del psi),

where del · is the divergence, del is the gradient, del ^2 is the Laplacian, and a·b is the dot product. From the divergence theorem,

 int_V(del ·F)dV=int_SF·da.

Plugging (2) into (3),

 int_Sphi(del psi)·da=int_V[phidel ^2psi+(del phi)·(del psi)]dV.

This is Green's first identity.

Subtracting (2) from (1),

 del ·(phidel psi-psidel phi)=phidel ^2psi-psidel ^2phi.


 int_V(phidel ^2psi-psidel ^2phi)dV=int_S(phidel psi-psidel phi)·da.

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


(Kaplan 1991, p. 361).

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

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Green's Identities

Cite this as:

Weisstein, Eric W. "Green's Identities." From MathWorld--A Wolfram Web Resource.

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