A quiet place behind the experiments

Keep the types
and the lineage in view.

The live page is for choosing fields and watching labels. This notebook is where the slower questions live: what kind of object is each symbol, what does a mesh actually store, why is the representation nonunique, and how does this little construction relate to fluid methods in the literature?

(\phi,\alpha,\beta)\in\Omega^0(S)^3\longrightarrow \eta=d\phi+\alpha d\beta\in\Omega^1(S)\longrightarrow\omega=d\eta\in\Omega^2(S)

Read straight through once; return later as a solver checklist.

Exterior calculus in six moves

What each object does.

The organizing principle is integration: zero-forms are sampled at points, one-forms are integrated along curves, and two-forms are integrated over oriented patches.

VECTOR ↔ COVECTOR

Motion and measurement are different jobs.

A tangent vector is a possible displacement. A covector consumes that displacement and returns a signed number. The metric relates the two, but does not make their roles identical.

THE METRIC DUAL

\eta=u^\flat,\qquad u=\eta^{\sharp_g},\qquad \eta(v)=g(u,v)

Advection uses u. Circulation integrates η.

LOOPS AND PATCHES

Circulation is not vorticity.

Circulation belongs to one selected loop. Vorticity is a local areal density. Stokes relates them only when the loop is the boundary of a patch.

STOKES

\Gamma(C)=\oint_C\eta,\qquad \omega^{(2)}=d\eta,\qquad \oint_{\partial A}\eta=\int_A\omega^{(2)}

\zeta=\star_g\omega^{(2)},\qquad \boldsymbol\omega=\zeta\,\mathbf n\ \text{for an oriented embedding}

The two-form is intrinsic. The scalar uses the surface metric and orientation; the normal vector additionally uses an embedding. A noncontractible torus loop is not ∂A and therefore carries additional period data.

CLEBSCH ASSEMBLY

Three scalar fields encode one one-form.

The term dφ is exact and curl-free. The weighted label differential αdβ supplies the part whose derivative is the wedge of two label gradients.

THE IDENTITY

\eta_C=d\phi+\alpha d\beta,\qquad d\eta_C=d\alpha\wedge d\beta

The identity d² = 0 removes d(dφ) and d(dβ).

INCOMPRESSIBLE PROJECTION

Subtract only the exact source–sink component.

Solve one scalar Poisson equation and subtract dp. This changes divergence but cannot change vorticity or a closed-loop period.

THE PRESERVATION AUDIT

\delta dp=\delta\eta_C,\qquad \eta_\perp=\eta_C-dp,\qquad d\eta_\perp=d\eta_C,\qquad \oint_C\eta_\perp=\oint_C\eta_C

This is the incompressible kinematic projection, not a shallow-water time step.

TOPOLOGY

Vorticity reconstruction leaves harmonic state.

On a closed genus-g surface, 2g noncontractible periods remain after local vorticity and divergence are fixed.

HODGE DECOMPOSITION

\eta=d\chi+\star_g d\psi+\sum_{i=1}^{2g}c_i h_i,\qquad \dim\mathcal H^1(S)=2g

For a torus, store or evolve two coefficients.

CLEBSCH GAUGE

The potentials are not unique physical observables.

Different triples can encode the same one-form. This representational freedom is distinct from changing a harmonic period, which changes the fluid.

ONE GAUGE FAMILY

\phi'=\phi-F(\beta),\qquad \alpha'=\alpha+F'(\beta),\qquad d\phi'+\alpha'd\beta=d\phi+\alpha d\beta

Gauge changes leave η fixed; harmonic subtraction does not.

Smooth object first · cochain second

What a triangle mesh stores.

DEC stores integrals of smooth forms on oriented mesh elements. The arrows shown in a viewer are reconstructed visualizations, not the primary discrete one-form.

01 · VERTICES

Sample the scalar labels

\alpha_i=\alpha(v_i),\qquad \beta_i=\beta(v_i),\qquad \phi_i=\phi(v_i)

Each vertex stores one number for each scalar field.

02 · EDGES

Differentiate by oriented differences

\int_{e_{ij}}d\alpha=\alpha_j-\alpha_i

Reversing the edge orientation reverses the sign.

03 · FACES

Store integrated vorticity

\int_f\omega=\oint_{\partial f}\eta

The signed face value is the circulation around its oriented boundary.

04 · METRIC

Introduce Hodge stars deliberately

\star_k:\mathcal C^k\longrightarrow\mathcal C^{2-k}

Incidence gives d; lengths, areas, and angles enter through the discrete Hodge stars.

05 · PROJECTION

Solve the scalar Poisson system

\delta d p=\delta\eta_C,\qquad \eta_\perp=\eta_C-dp

Audit divergence, face circulation, and harmonic periods separately.

Return to an experiment

Close the notebook when the picture feels clear again.

The flat-torus lab isolates the two harmonic coefficients. The shallow-water lab shows why divergence becomes part of the dynamics rather than something to erase.

NEXT EXPERIMENTCompare two harmonic periods with material grids →