In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential operators act. They are named after the mathematician George Green, who discovered Green's theorem. Green's first identity This identity is derived from the divergence theorem applied to the vector field while using an extension of the product rule that : Let and be scalar functions defined on some region , and suppose that is twice continuously differentiable, and is once continuously differentiable. Using the product rule above, but letting , integrate over . Then where is the Laplace operator, is the boundary of region , is the outward pointing unit normal to the surface element and is the oriented surface element. This theorem is a special case of the divergence theorem, and is essentially the higher dimensional equivalent of integration by parts with and the gradient of replacing and . Note that Green's first identity above is a special case of the more general identity derived from the divergence theorem by substituting , Green's second identity If and are both twice continuously differentiable on , and is once continuously differentiable, one may choose to obtain For the special case of all across , then, In the equation above, is the directional derivative of in the direction of the outward pointing surface normal of the surface element , Explicitly incorporating this definition in the Green's second identity with results in In particular, this demonstrates that the Laplacian is a self-adjoint operator in the inner product for functions vanishing on the boundary so that the right hand side of the above identity is zero. Green's third identity Green's third identity derives from the second identity by choosing , where the Green's function is taken to be a fundamental solution of the Laplace operator, ∆. This means that: For example, in , a solution has the form Green's third identity states that if is a function that is twice continuously differentiable on , then A simplification arises if is itself a harmonic function, i.e. a solution to the Laplace equation. Then and the identity simplifies to The second term in the integral above can be eliminated if is chosen to be the Green's function that vanishes on the boundary of (Dirichlet boundary condition), This form is used to construct solutions to Dirichlet boundary condition problems. Solutions for Neumann