**Green’s Identities**
**Definition**
Green’s identities are a set of integral formulas in vector calculus that relate the integrals of functions and their derivatives over a volume to integrals over the boundary surface of the volume. They are fundamental tools in mathematical physics and partial differential equations, particularly in solving boundary value problems.
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# Green’s Identities
Green’s identities are a collection of three integral identities that play a central role in vector calculus, potential theory, and the theory of partial differential equations (PDEs). Named after the British mathematician George Green, who introduced them in the early 19th century, these identities establish relationships between functions and their Laplacians within a domain and on its boundary. They are instrumental in deriving solutions to boundary value problems, formulating weak solutions, and proving uniqueness and existence theorems in mathematical physics and engineering.
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## Historical Background
George Green first published these integral formulas in his 1828 essay, „An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism.” His work laid the foundation for modern potential theory and influenced the development of mathematical physics. Although initially overlooked, Green’s identities gained prominence in the late 19th and early 20th centuries as the theory of PDEs and boundary value problems advanced.
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## Mathematical Preliminaries
Before stating Green’s identities, it is essential to establish the mathematical context and notation.
– **Domain and Boundary:** Let ( Omega subset mathbb{R}^n ) be a bounded, open, and sufficiently smooth domain with boundary ( partial Omega ).
– **Functions:** Consider scalar functions ( u, v in C^2(overline{Omega}) ), meaning they are twice continuously differentiable on the closure of ( Omega ).
– **Gradient and Laplacian:** The gradient of a function ( u ) is denoted by ( nabla u ), and the Laplacian by ( Delta u = nabla cdot nabla u ).
– **Normal Vector:** The outward unit normal vector to the boundary ( partial Omega ) is denoted by ( mathbf{n} ).
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## Statement of Green’s Identities
Green’s identities relate volume integrals over ( Omega ) to surface integrals over ( partial Omega ). There are three classical identities, often referred to as the first, second, and third Green’s identities.
### First Green’s Identity
The first Green’s identity states that for functions ( u, v in C^2(overline{Omega}) ),
[
int_{Omega} left( u Delta v + nabla u cdot nabla v right) , dV = int_{partial Omega} u frac{partial v}{partial n} , dS,
]
where ( frac{partial v}{partial n} = nabla v cdot mathbf{n} ) is the normal derivative of ( v ) on the boundary.
**Interpretation:** This identity expresses a relationship between the Laplacian of ( v ) weighted by ( u ), the inner product of gradients of ( u ) and ( v ), and the boundary flux of ( v ) weighted by ( u ).
### Second Green’s Identity
The second Green’s identity is derived by interchanging the roles of ( u ) and ( v ) in the first identity and subtracting:
[
int_{Omega} left( u Delta v – v Delta u right) , dV = int_{partial Omega} left( u frac{partial v}{partial n} – v frac{partial u}{partial n} right) , dS.
]
**Interpretation:** This identity relates the difference of Laplacians weighted by the other function to the difference of their normal derivatives on the boundary.
### Third Green’s Identity
The third Green’s identity involves the fundamental solution (or Green’s function) ( G(x, y) ) of the Laplacian operator in ( Omega ). For a fixed point ( y in Omega ), and ( u in C^2(overline{Omega}) ), it states:
[
u(y) = int_{partial Omega} left( G(x, y) frac{partial u}{partial n}(x) – u(x) frac{partial G}{partial n_x}(x, y) right) dS_x + int_{Omega} G(x, y) Delta u(x) , dV_x,
]
where ( frac{partial}{partial n_x} ) denotes differentiation with respect to the outward normal at ( x ).
**Interpretation:** This identity expresses the value of ( u ) at an interior point ( y ) in terms of boundary integrals involving ( u ) and its normal derivative, and a volume integral involving the Laplacian of ( u ). It is fundamental in potential theory and boundary integral methods.
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## Derivation of Green’s Identities
The derivations of Green’s identities rely on the divergence theorem (also known as Gauss’s theorem) and properties of the gradient and Laplacian operators.
### Divergence Theorem
The divergence theorem states that for a vector field ( mathbf{F} in C^1(overline{Omega}, mathbb{R}^n) ),
[
int_{Omega} nabla cdot mathbf{F} , dV = int_{partial Omega} mathbf{F} cdot mathbf{n} , dS.
]
This theorem converts volume integrals of divergences into surface integrals of fluxes.
### Derivation of the First Green’s Identity
Consider the vector field ( mathbf{F} = u nabla v ). Then,
[
nabla cdot (u nabla v) = nabla u cdot nabla v + u Delta v.
]
Applying the divergence theorem,
[
int_{Omega} nabla cdot (u nabla v) , dV = int_{partial Omega} u nabla v cdot mathbf{n} , dS.
]
Substituting the divergence expression,
[
int_{Omega} left( nabla u cdot nabla v + u Delta v right) dV = int_{partial Omega} u frac{partial v}{partial n} , dS,
]
which is the first Green’s identity.
### Derivation of the Second Green’s Identity
Apply the first Green’s identity twice, once with ( (u, v) ) and once with ( (v, u) ):
[
int_{Omega} left( u Delta v + nabla u cdot nabla v right) dV = int_{partial Omega} u frac{partial v}{partial n} dS,
]
[
int_{Omega} left( v Delta u + nabla v cdot nabla u right) dV = int_{partial Omega} v frac{partial u}{partial n} dS.
]
Subtracting the second from the first,
[
int_{Omega} left( u Delta v – v Delta u right) dV = int_{partial Omega} left( u frac{partial v}{partial n} – v frac{partial u}{partial n} right) dS,
]
which is the second Green’s identity.
### Derivation of the Third Green’s Identity
The third identity uses the second identity with ( v = G(cdot, y) ), the Green’s function for the Laplacian in ( Omega ) with singularity at ( y ). Since ( Delta_x G(x, y) = -delta(x – y) ), where ( delta ) is the Dirac delta distribution, substituting into the second identity and rearranging yields the third Green’s identity.
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## Applications of Green’s Identities
Green’s identities have broad applications in mathematics, physics, and engineering, particularly in the study of PDEs and potential theory.
### Boundary Value Problems
Green’s identities are fundamental in solving boundary value problems for elliptic PDEs such as Laplace’s equation, Poisson’s equation, and Helmholtz equation. They allow the transformation of PDEs into integral equations on the boundary, facilitating analytical and numerical solutions.
### Potential Theory
In potential theory, Green’s identities help express potentials in terms of boundary data. The third Green’s identity, involving the Green’s function, is especially important for representing harmonic functions and constructing solutions with prescribed boundary conditions.
### Weak Formulations and Variational Methods
Green’s first identity is used to derive weak formulations of PDEs, which are essential in the finite element method and other numerical techniques. By integrating by parts, derivatives are transferred from one function to another, enabling the use of less regular function spaces.
### Uniqueness and Existence Theorems
The second Green’s identity is instrumental in proving uniqueness theorems for boundary value problems. For example, it can be used to show that the solution to the Dirichlet or Neumann problem for Laplace’s equation is unique under appropriate conditions.
### Electromagnetism and Fluid Mechanics
Green’s identities underpin many integral equation methods in electromagnetism and fluid mechanics, where fields satisfy Laplace or Poisson equations. They facilitate the derivation of integral representations of fields and potentials.
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## Generalizations and Related Identities
Green’s identities have been generalized and extended in various directions.
### Green’s Formulas for Vector Fields
Analogous identities exist for vector fields involving the divergence and curl operators, such as the vector Green’s identities used in electromagnetism and elasticity.
### Green’s Identities in Riemannian Manifolds
On Riemannian manifolds, Green’s identities generalize to relate integrals involving the Laplace-Beltrami operator and boundary integrals, incorporating the manifold’s metric and curvature.
### Green’s Identities for Other Differential Operators
Similar integral identities exist for other elliptic operators, such as the biharmonic operator, enabling the study of higher-order PDEs.
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## Examples
### Example 1: Laplace’s Equation with Dirichlet Boundary Conditions
Let ( u ) be harmonic in ( Omega ) (i.e., ( Delta u = 0 )) with prescribed values on ( partial Omega ). Using the third Green’s identity with the Green’s function ( G ), the solution can be represented as
[
u(y) = int_{partial Omega} left( G(x, y) frac{partial u}{partial n}(x) – u(x) frac{partial G}{partial n_x}(x, y) right) dS_x.
]
If ( u ) is known on the boundary, this formula expresses ( u ) inside ( Omega ) in terms of boundary data.
### Example 2: Weak Formulation of Poisson’s Equation
Consider Poisson’s equation,
[
-Delta u = f quad text{in } Omega,
]
with Dirichlet boundary conditions ( u = 0 ) on ( partial Omega ). Multiplying by a test function ( v ) and integrating over ( Omega ), Green’s first identity gives
[
int_{Omega} nabla u cdot nabla v , dV = int_{Omega} f v , dV,
]
which is the weak formulation used in finite element methods.
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## Summary
Green’s identities are foundational integral relations in vector calculus that connect the behavior of functions and their derivatives inside a domain to their behavior on the boundary. They provide essential tools for solving and analyzing partial differential equations, especially those arising in physics and engineering. Their derivation from the divergence theorem and their applications in boundary value problems, potential theory, and numerical methods underscore their central role in mathematical analysis.
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## See Also
– Divergence Theorem
– Laplace’s Equation
– Green’s Function
– Boundary Value Problems
– Potential Theory
– Finite Element Method
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## Meta Description
Green’s identities are integral formulas in vector calculus relating functions and their derivatives over a domain to integrals on its boundary, fundamental in solving partial differential equations and boundary value problems.