Regular value: Difference between revisions
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Given a [[smooth map]] of [[differential manifold]]s, a point in the image manifold is termed a '''regular value''' for the smooth map if ''every'' point in its inverse image is a [[regular point]], i.e. if the map from the tangent space at any point in the inverse image, is surjective. | Given a [[smooth map]] of [[differential manifold]]s, a point in the image manifold is termed a '''regular value''' for the smooth map if ''every'' point in its inverse image is a [[regular point]], i.e. if the map from the tangent space at any point in the inverse image, is surjective. | ||
Note that any point whose inverse image is empty, is by definition a regular value. | |||
===Definition with symbols=== | ===Definition with symbols=== | ||
Let <math>f:M \to N</math> be a smooth map of differential manifolds. A point <math>p \in N</math> is termed a '''regular value''' of <math>f</math> if for every <math>m \in f^{-1}(\{p\})</math>, the induced map <math>(df)_m: T_mM \to T_pN</math>, is surjective. | Let <math>f:M \to N</math> be a smooth map of differential manifolds. A point <math>p \in N</math> is termed a '''regular value''' of <math>f</math> if for every <math>m \in f^{-1}(\{p\})</math>, the induced map <math>(df)_m: T_mM \to T_pN</math>, is surjective. | ||
Revision as of 00:39, 13 January 2008
Definition
Symbol-free definition
Given a smooth map of differential manifolds, a point in the image manifold is termed a regular value for the smooth map if every point in its inverse image is a regular point, i.e. if the map from the tangent space at any point in the inverse image, is surjective.
Note that any point whose inverse image is empty, is by definition a regular value.
Definition with symbols
Let be a smooth map of differential manifolds. A point is termed a regular value of if for every , the induced map , is surjective.
