Hodge Laplacian
Helmholtz Decomposition of 2D Vector Fields
Vector field (click to add source/sink)
Decomposition: curl-free | div-free | harmonic
Display
Original F
Curl-free ∇φ
Div-free ∇×A
Harmonic part
div(F) heatmap
curl(F) heatmap
Field type
Vortex
Source
Saddle
Uniform
Shear
Mixed
Arrow scale
1.0
Grid resolution
12
Rebuild
Clear Sources
Animate flow
Helmholtz theorem: any vector field F = ∇φ (curl-free) + ∇×A (div-free) + harmonic.
∇·(∇φ) = ∇·F → solve Poisson for φ. ∇²A = −∇×F → solve for stream function.
Hodge decomposition generalizes this to differential forms on any manifold.