A reference page, not another experiment

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in view.

The live page is now reserved for choosing fields and watching material labels. This page holds the exterior-calculus bookkeeping you need when implementing the same ideas on a mesh.

(\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

Use the reference only when a type or operator becomes ambiguous.

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

NEXT EXPERIMENTCompare two harmonic periods with material grids →

Keep the typesin view.