188 Smooth Mapping
A smooth mapping is a mapping of a vector to another vector where every derivative of the mapping satisfies continuity that is used to calculate a change in a variable value that satisfies continuity.
Note: Also called smooth map. Also called \(C^{\infty}\) map.
definition [d] (Smooth Mapping = Smooth Map) From Kosinski: let \(f: M \rightarrow N\) where \(M\), \(N\) are differential manifolds. The mapping \(f\) is smooth if there are atlases \(\{U_{\alpha}, h_{\alpha}\}\) on \(M\) and \(\{V_{\beta}, g_{\beta}\}\) on \(N\) such that the maps
- \(g_{\beta} \circ f \circ h_{\alpha}^{-1}\)
are smooth wherever they are defined. The mapping \(f\) is a diffeomorphism if it is smooth and has a smooth inverse.
where
- \(M\), \(N\) are differential manifolds.
- \(f\) is the mapping being tested for smoothness.
- \(\{U_{\alpha}, h_{\alpha}\}\) is an atlas on \(M\).
- \(\{V_{\beta}, g_{\beta}\}\) is an atlas on \(N\).
definition [d] (Smooth Mapping = \(C^{\infty}\) Map) From Tu: let \(N\) and \(M\) be manifolds of dimension \(n\) and \(m\) respectively. A continuous map \(F: N \rightarrow M\) is \(C^{\infty}\) at a point \(p\) in \(N\) if there are charts \((V,\psi)\) about \(F(p)\) in \(M\) and \((U,\phi)\) about \(p\) in \(N\) such that the composition
- \(\psi \circ F \circ \phi^{-1}\)
a map from the open subset \(\phi\bigl(F^{-1}(V) \cap U\bigr)\) of \(\mathbb{R}^{n}\) to \(\mathbb{R}^{m}\), is \(C^{\infty}\) at \(\phi(p)\). The continuous map \(F: N \rightarrow M\) is said to be \(C^{\infty}\) if it is \(C^{\infty}\) at every point of \(N\).
where
- \(F\) is the continuous map.
- \((U,\phi)\) is a chart about \(p\) in \(N\).
- \((V,\psi)\) is a chart about \(F(p)\) in \(M\).
- \(\psi \circ F \circ \phi^{-1}\) is the local Euclidean representative.
- \(C^{\infty}\) means infinitely differentiable.
188.1 Elementary Example
188.1.1 Simple
A smooth mapping maps each domain point to a unique image. A finite sample of \(f(x) = x^{2}\) shows the rule.
\[ f : A \rightarrow B \]
\[ A = \{ 0,\ 1,\ 2,\ 3 \} \]
\[ B = \{ 0,\ 1,\ 4,\ 9 \} \]
\[ f(0) = 0,\quad f(1) = 1,\quad f(2) = 4,\quad f(3) = 9 \]
where
- \(f\) is the mapping.
- \(A\) is the domain sample.
- \(B\) is the image sample.
188.1.2 General
A smooth map \(\mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) has a Jacobian matrix of first partial derivatives at each point.
\[ f(x,y) = (x^{2},\, xy) \]
\[ Df = \begin{pmatrix} 2x & 0 \\ y & x \end{pmatrix} \]
where
- \(Df\) is the Jacobian matrix of \(f\).
- smoothness means all partial derivatives of all orders exist and are continuous.
188.2 References
- Kosinski, A. A. Differential Manifolds. — Definition (1.6): smooth maps via \(g_{\beta} \circ f \circ h_{\alpha}^{-1}\); diffeomorphism.
- Tu, L. W. An Introduction to Manifolds. — Definition 6.5: \(C^{\infty}\) at a point via \(\psi \circ F \circ \phi^{-1}\).
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