Difference between revisions of "Smooth map"
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A map {{M|f:M\rightarrow N}} between two [[Smooth manifold|smooth manifolds]] {{M|(M,\mathcal{A})}} and {{M|(N,\mathcal{B})}} (of not necessarily the same dimension) is said to be smooth<ref>Introduction to smooth manifolds - John M Lee - Second Edition</ref> if: | A map {{M|f:M\rightarrow N}} between two [[Smooth manifold|smooth manifolds]] {{M|(M,\mathcal{A})}} and {{M|(N,\mathcal{B})}} (of not necessarily the same dimension) is said to be smooth<ref>Introduction to smooth manifolds - John M Lee - Second Edition</ref> if: | ||
* <math>\forall p\in M\exists\ (U,\varphi)\in\mathcal{A},\ p\in U\text{ and }(V,\psi)\in\mathcal{B}</math> such that <math>F(U)\subseteq V\wedge[\psi\circ F\circ\varphi^{-1}:\varphi(U)\rightarrow\psi(V)]</math> is [[Smooth|smooth]] | * <math>\forall p\in M\exists\ (U,\varphi)\in\mathcal{A},\ p\in U\text{ and }(V,\psi)\in\mathcal{B}</math> such that <math>F(U)\subseteq V\wedge[\psi\circ F\circ\varphi^{-1}:\varphi(U)\rightarrow\psi(V)]</math> is [[Smooth|smooth]] | ||
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===Via commutative diagrams=== | ===Via commutative diagrams=== | ||
A map is smooth if the following diagram commutes: | A map is smooth if the following diagram commutes: | ||
Latest revision as of 21:37, 14 April 2015
Note: not to be confused with smooth function
Definition
A map f:M→N between two smooth manifolds (M,A) and (N,B) (of not necessarily the same dimension) is said to be smooth[1] if:
- ∀p∈M∃ (U,φ)∈A, p∈U and (V,ψ)∈Bsuch that F(U)⊆V∧[ψ∘F∘φ−1:φ(U)→ψ(V)]is smooth
See also
References
- Jump up ↑ Introduction to smooth manifolds - John M Lee - Second Edition
