Exterior Forms and the Exterior Derivative

ExteriorDerivative encodes the topological structure of the mesh as sparse incidence matrices (vertex-to-edge) and (edge-to-triangle). These are the discrete analogues of the continuous exterior derivative .

Summary

Key property (exactness): ; the boundary of a boundary is zero.

Code Examples

Building the exterior derivative

use cartan_dec::{Mesh, ExteriorDerivative};
 
let mesh = Mesh::unit_square_grid(4);
let ext  = ExteriorDerivative::from_mesh(&mesh);

Applying : discrete gradient of a scalar field

use nalgebra::DVector;
 
// Scalar field f on vertices
let f = DVector::from_fn(mesh.n_vertices(), |i, _| {
    let [x, y] = mesh.vertices[i];
    x + y
});
 
// df: 1-form on edges - direct matrix multiplication
let df = &ext.d0 * &f;
assert_eq!(df.len(), mesh.n_edges());

Applying : discrete curl of a 1-form

// curl(df) = d₁(d₀(f)) = 0 for exact 1-forms
let curl_df = &ext.d1 * &df;
assert!(curl_df.norm() < 1e-12);  // exactness: d∘d = 0

Formula Reference

incidence: for edge with :

incidence: for triangle and edge : +1 if appears positively oriented in , -1 if negatively, 0 if absent.

Numerical Notes

Sparsity: has exactly non-zeros; has exactly . Both are represented as sparse matrices for efficient matrix-vector products.

Exactness: is exact by construction (not approximate). This is a topological identity that holds at machine precision.

Applications

• Discrete Hodge decomposition:
• Building Laplacians:
• Checking closedness/exactness of discrete differential forms