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Canonical static concept record

Exterior derivative

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Summary

The canonical linear operator $d:\Omega^k(M)\to\Omega^{k+1}(M)$ extending the differential of smooth functions, satisfying $d^2=0$ and the graded Leibniz rule.

Record metadata

Canonical name: Exterior derivative
Concept ID: exterior_derivative
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Differential geometry
Domains: Differential geometry, Analysis
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Differential-form operator

Concept sources

Lee — Introduction to Smooth Manifolds — Introduction to Smooth Manifolds · textbook · source ID lee-smooth-manifolds
Lee — Introduction to Riemannian Manifolds — Introduction to Riemannian Manifolds · textbook · source ID lee-riemannian-manifolds

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Smooth manifold Exterior derivative

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: differential_geometry_smooth_manifold_to_exterior_derivative
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: define coordinate-independent differentiation

Authored explanation

Smooth coordinate changes make the exterior derivative of differential forms globally well defined, independently of the local chart used to calculate it.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Lee — Introduction to Smooth Manifolds — Introduction to Smooth Manifolds · textbook · source ID lee-smooth-manifolds

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