Green's Function for the 3D Laplacian
This animation builds the Green's function for the negative Laplacian in 3D using a point electric charge as physical intuition. Starting from a 3D coordinate system with observation point r and source point r′, it introduces the Dirac delta function as an idealized point-charge density, verifies that integrating it recovers total charge q, and connects this to the equation defining the Green's function. Useful for students in electromagnetism or PDE courses linking physical charge distributions to mathematical source terms.
16:9 · every frame verified for overlaps, spacing and edges before rendering
The prompt that made it
Create a visually intuitive educational math animation explaining the Green’s function for the 3D negative Laplacian, using a point electric charge as the physical interpretation. The goal is NOT just to display equations. The animation should build the ideas visually from simple concepts to the final Green’s-function relationship. ### Scene 1 — Set up 3D space Show a clean 3D coordinate system with x, y, and z axes. Place a single point at position r′ and label it “point charge q”. Show a position vector from the origin to the charge labeled r′. Then introduce a second position vector r representing an arbitrary observation point in space. Visually distinguish r and r′. ### Scene 2 — Introduce the delta function Show the point charge as a sharply localized spike at r′. Display ρ(r) = q δ³(r − r′) Animate the delta function as becoming increasingly narrow and tall while its total “area/volume” remains associated with q. Explain visually that δ³(r − r′) is zero everywhere except at r = r′ and represents a point source. Avoid pretending that the delta function is literally an ordinary finite-height spike; indicate that the spike is a visual representation. ### Scene 3 — Verify the total charge Show the charge density distributed over 3D space and then display ∫ d³r ρ(r) = q Substitute the density: ∫ d³r q δ³(r − r′) Then visually show the delta function selecting the point r = r′: = q The animation should emphasize: “Integrating a point-charge density over ALL space gives the total charge q.” ### Scene 4 — Connect charge density to potential Introduce the electric potential φ(r) as a scalar field surrounding the point charge. Show concentric spherical equipotential surfaces around the charge, with the potential becoming weaker farther from the charge. Then display the Poisson equation: −∇² ε₀ φ(r) = q δ³(r − r′) Visually explain that the Laplacian describes how the potential field responds to the localized source. ### Scene 5 — Introduce the Green function Replace the physical charge q with an abstract mathematical point source. Display the Green function equation: −∇² G(r,r′) = δ³(r − r′) Animate the transition: point source → Green function → resulting spatial field Make it clear that G(r,r′) is the response of the operator −∇² to a point source located at r′. ### Scene 6 — Make the physical interpretation Return to the electric charge. Display: ε₀ φ(r) = q G(r,r′) Then visually show that the Green function is essentially the potential produced by a unit point source, up to the factor q/ε₀. Use a simple side-by-side conceptual transformation: δ³(r − r′) ↓ −∇² ↓ G(r,r′) and q δ³(r − r′) ↓ Poisson equation ↓ ε₀ φ(r) ### Scene 7 — Final takeaway End with the point charge in the center and spherical potential field surrounding it. Show the three key equations together, appearing one at a time: ρ(r) = qδ³(r − r′) −∇²G(r,r′) = δ³(r − r′) ε₀φ(r) = qG(r,r′) Finish with the conceptual statement: “A Green function tells us the response of an operator to a point source.” Use smooth animations, arrows, spatial transformations, and color/opacity changes to distinguish the source, the observation point, the potential field, and the mathematical Green function. Keep the pacing slow enough for a student learning the material for the first time. Every equation should appear only after the visual concept it represents has been introduced.
Make the next one in this series
- Derive the explicit form G(r,r') = 1/(4π|r−r'|) by solving the Poisson equation directly.
- Show how superposition of many point charges builds the general solution via Green's function integration.
- Visualize the electric field and potential surrounding a point charge in 3D space.
