Green's identities are a set of three vector derivative/integral identities which can be derived starting with the vector derivative identities
is the divergence, is the gradient, is the Laplacian, and is the dot
product. From the divergence theorem,
Plugging (2) into (3),
This is Green's first identity.
Subtracting (2) from (1),
This is Green's second identity.
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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ReferencesKaplan, W. Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, 1991.
on Wolfram|AlphaGreen's Identities
Cite this as:
Weisstein, Eric W. "Green's Identities."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GreensIdentities.html