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Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

Smooth curve

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Summary

A smooth map from an interval or one-dimensional manifold into a smooth manifold.

Record metadata

Canonical name: Smooth curve
Concept ID: smooth_curve
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: Smooth map

Carrier(s)

interval or one-dimensional smooth manifold $I$
smooth manifold $M$

Data

smooth map $\gamma:I\to M$

Axioms / constraints

smoothness

Canonically induces

tangent vector $\dot\gamma(t)$
pullback along the curve

Concept sources

Wikipedia — Curve (geometry) — Curve (geometry) · encyclopedic reference · source ID wp-smooth_curve
nLab — smooth curve — smooth curve · mathematical reference · source ID nlab-smooth_curve
Lee — Introduction to Smooth Manifolds — Introduction to Smooth Manifolds · textbook · source ID lee-smooth-manifolds

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Smooth manifold Smooth curve

Permalink to relation

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

Relation ID: smoothcurve_from_manifold
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: smooth parameterized map

Authored explanation

A smooth curve is a smooth map from a one-dimensional parameter domain into a smooth manifold.

How to interpret this relation type

Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.

Relation sources

Wikipedia — Curve (geometry) — Curve (geometry) · encyclopedic reference · source ID wp-smooth_curve
nLab — smooth curve — smooth curve · mathematical reference · source ID nlab-smooth_curve
Lee — Introduction to Smooth Manifolds — Introduction to Smooth Manifolds · textbook · source ID lee-smooth-manifolds

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

Smooth curve Tangent vector

Permalink to relation

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

Relation ID: tangentvector_from_curve
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: curve velocity

Authored explanation

The derivative of a smooth curve at a parameter value defines a tangent vector.

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

Wikipedia — Tangent space — Tangent space · encyclopedic reference · source ID wp-tangent_vector
nLab — tangent vector — tangent vector · mathematical reference · source ID nlab-tangent_vector
Lee — Introduction to Smooth Manifolds — Introduction to Smooth Manifolds · textbook · source ID lee-smooth-manifolds

Smooth curve Worldline

Permalink to relation

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

Relation ID: smoothcurve_to_worldline
Relation type: Mathematical formulation mathematical-formulation
Direction: source → target
Endpoint roles: source: Mathematically formulates; target: Mathematically formulated using
Authored annotation: smooth spacetime curve

Authored explanation

Ideal particle histories are represented by smooth causal curves in spacetime.

How to interpret this relation type

A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.

Relation sources

Wikipedia — Curve (geometry) — Curve (geometry) · encyclopedic reference · source ID wp-smooth_curve
nLab — smooth curve — smooth curve · mathematical reference · source ID nlab-smooth_curve
Lee — Introduction to Smooth Manifolds — Introduction to Smooth Manifolds · textbook · source ID lee-smooth-manifolds
Wikipedia — World line — World line · encyclopedic reference · source ID wp-physics-worldline
nLab — worldline — worldline · mathematical physics reference · source ID nlab-physics-worldline
Einstein Online — Special Relativity: Spacetime — Special Relativity: Spacetime · Max Planck Institute reference · source ID einstein-spacetime
Cambridge — Dynamics and Relativity — Dynamics and Relativity · University of Cambridge lecture notes · source ID tong-relativity

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Smooth curve

Summary

Record metadata

Carrier(s)

Data

Axioms / constraints

Canonically induces

Concept sources

Incoming relations (arrows to this concept)

Smooth manifold Smooth curve

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)

Smooth curve Tangent vector

Authored explanation

How to interpret this relation type

Relation sources

Smooth curve Worldline

Authored explanation

How to interpret this relation type

Relation sources