one number at every vertex
THE NEXT RUNG
Vertex scalar, edge 1-form, face vector, or vertex tangent vector are different spaces.
d₀φ is an edge difference. d₁α is triangle circulation.
d₀φ
d₁α
The Hodge star supplies geometry and units; it is not another incidence matrix.
TinyAD differentiates the edge or face term and scatters its sparse derivatives.