Global analysis, analysis on manifolds Concept
Differential of a smooth map
Summary
The differential of a smooth map between manifolds is its best linear approximation at a point. It sends tangent vectors at a point of the source to tangent vectors at the image point in the target. In local coordinates it is given by the Jacobian matrix of partial derivatives. It extends the derivative of calculus to curved spaces, and it obeys the chain rule.
Where this leads
21 pages build on this one.
- Morse–Smale system
- Left-invariant vector field
- Adjoint representation of a Lie group
- Lagrangian singularity
- Thom–Mather isotopy lemmas
- Smale–Hirsch theorem
- Isometry of Riemannian manifolds
- Symmetric space
- Infinitesimal stability
- Transversality
- Submersion
- Whitney umbrella
- Regular value theorem
- Singularity of a smooth map
- Tautological one-form
- Thom–Mather stratified space
- Ehresmann connection
- Maurer–Cartan form
- Stability conjecture
- Immersion
- Thom–Boardman singularities
Further reading
About this page
| Tag | 6P9PXQDSEN |
|---|---|
| URI | m2://6P9PXQDSEN |
| Kind | concept |
| Writing | W0 stub |
| MSC | 58A05, 57R35 |
| Same as |
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Used by
- Left-invariant vector field
- Adjoint representation of a Lie group
- Lagrangian singularity
- Smale–Hirsch theorem
- Isometry of Riemannian manifolds
- Symmetric space
- Infinitesimal stability
- Transversality
- Submersion
- Whitney umbrella
- Regular value theorem
- Singularity of a smooth map
- Tautological one-form
- Ehresmann connection
- Maurer–Cartan form
- Immersion
- Thom–Boardman singularities