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Green's Identities — Definition, Formula & Examples

Green's identities are a set of integral relations that connect volume integrals involving the Laplacian and gradient of two scalar functions to surface integrals over the boundary. They are derived by applying the divergence theorem to specific vector fields built from those scalar functions.

Let uu and vv be scalar fields that are twice continuously differentiable on a bounded region V⊂R3V \subset \mathbb{R}^3 with piecewise-smooth boundary ∂V\partial V and outward unit normal n^\hat{n}. Green's first identity states ∫V(u ∇2v+∇u⋅∇v) dV=∮∂Vu ∂v∂n dS\int_V (u\,\nabla^2 v + \nabla u \cdot \nabla v)\,dV = \oint_{\partial V} u\,\frac{\partial v}{\partial n}\,dS. Green's second identity (the symmetric form) states ∫V(u ∇2v−v ∇2u) dV=∮∂V ⁣(u ∂v∂n−v ∂u∂n)dS\int_V (u\,\nabla^2 v - v\,\nabla^2 u)\,dV = \oint_{\partial V}\!\left(u\,\frac{\partial v}{\partial n} - v\,\frac{\partial u}{\partial n}\right)dS.

Key Formula

∫V ⁣(u ∇2v+∇u⋅∇v) dV=∮∂Vu ∂v∂n dS\int_V \!\bigl(u\,\nabla^2 v + \nabla u \cdot \nabla v\bigr)\,dV = \oint_{\partial V} u\,\frac{\partial v}{\partial n}\,dS
Where:
  • u,vu, v = Twice continuously differentiable scalar fields on V
  • VV = Bounded region in ℝ³ with piecewise-smooth boundary
  • ∇2v\nabla^2 v = Laplacian of v
  • ∂v∂n\frac{\partial v}{\partial n} = Normal derivative of v on the boundary, i.e. ∇v · n̂
  • dSdS = Surface area element on ∂V

How It Works

Start with two scalar functions uu and vv defined on a volume VV. To obtain the first identity, apply the divergence theorem to the vector field u ∇vu\,\nabla v. Expanding ∇⋅(u ∇v)\nabla \cdot (u\,\nabla v) via the product rule gives u ∇2v+∇u⋅∇vu\,\nabla^2 v + \nabla u \cdot \nabla v, and the surface integral becomes ∮u (∇v⋅n^) dS\oint u\,(\nabla v \cdot \hat{n})\,dS. For the second identity, write the first identity with the roles of uu and vv swapped and subtract one from the other; the ∇u⋅∇v\nabla u \cdot \nabla v terms cancel, leaving a purely symmetric relation. These identities are the main tool for proving uniqueness of solutions to Laplace's and Poisson's equations.

Example

Problem: Let v be harmonic (∇²v = 0) on a volume V with boundary ∂V. Use Green's first identity with u = v to show that if v = 0 on ∂V, then v = 0 everywhere in V.
Step 1: Set u = v in Green's first identity and use ∇²v = 0.
∫V(v ∇2v⏟0+∣∇v∣2) dV=∮∂Vv⏟0 ∂v∂n dS\int_V \bigl(v\,\underbrace{\nabla^2 v}_{0} + |\nabla v|^2\bigr)\,dV = \oint_{\partial V} \underbrace{v}_{0}\,\frac{\partial v}{\partial n}\,dS
Step 2: Both the Laplacian term and the boundary term vanish, leaving:
∫V∣∇v∣2 dV=0\int_V |\nabla v|^2\,dV = 0
Step 3: Since |∇v|² ≥ 0 everywhere and its integral is zero, we conclude ∇v = 0 throughout V. Combined with v = 0 on the boundary, this forces v = 0 in all of V.
Answer: A harmonic function that vanishes on the boundary must be identically zero in the interior — proving uniqueness of the Dirichlet problem for Laplace's equation.

Why It Matters

Green's identities are essential in PDE theory for proving that solutions to Laplace's and Poisson's equations are unique. They also underpin the boundary element method used in engineering to solve heat conduction, electrostatics, and fluid flow problems numerically.

Common Mistakes

Mistake: Confusing Green's identities (3D volume/surface integrals) with Green's theorem (2D line/area integrals).
Correction: Green's theorem relates a line integral around a planar curve to a double integral over the enclosed region. Green's identities are higher-dimensional results derived from the divergence theorem, not from Green's theorem directly.