What is the Green's function for the Laplacian in three dimensions (free space)?
AG(x,y) = -1/(4π|x-y|)
BG(x,y) = -1/(2π) ln|x-y|
CG(x,y) = |x-y|²
DG(x,y) = e^(-|x-y|)
The free-space Green's function for -Δ in 3D is G(x,y) = 1/(4π|x-y|). This is the electrostatic potential due to a unit point charge, reflecting the inverse-square law in three dimensions.
Question 2 True / False
The Green's function for the Laplacian on a bounded domain depends on the shape of the domain.
TTrue
FFalse
Answer: True
The Green's function must satisfy boundary conditions (typically G = 0 on ∂Ω for Dirichlet problems), so it depends on the domain geometry. The free-space fundamental solution is domain-independent, but the Green's function adds a correction term to enforce the boundary condition.
Question 3 Short Answer
How does the Green's function relate to the superposition principle?
Think about your answer, then reveal below.
Model answer: It decomposes any source f into point sources and sums their individual responses via integration
For a linear operator, the response to an arbitrary source f is the integral (superposition) of responses to point sources δ(x-y) weighted by f(y). This is the essence of the representation u(x) = ∫G(x,y)f(y)dy.
Question 4 True / False
The Green's function G(x,y) for a self-adjoint operator satisfies G(x,y) = G(y,x).
TTrue
FFalse
Answer: True
This reciprocity property means the response at x due to a source at y equals the response at y due to a source at x. It follows from the self-adjointness of the operator and is physically meaningful: it expresses Maxwell's reciprocity principle in electrostatics.