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Unformatted text preview: 1.3. TANGENT VECTORS AND MAPPINGS 17 1.3.4 Mappings Let M be an ndimensional manifold and N be an mdimensional manifold and let F : M N be a map from M to N . Let ( U , ) A be an atlas in M and ( V , ) B be an atlas in N . We define the maps F = F  1 : ( U ) ( V ) from open sets in R n to R m defined by y a = F a ( x 1 , . . . , x n ) , where a = 1 , . . . , m . The map F is said to be smooth if F a are smooth functions of local coor dinates x i , i = 1 , . . . , n . The process of replacing the map F by the functions F = F  1 is usually omitted, and the maps F and F are identified, so that we think of the map F directly in terms of local coordinates. If n = m and the map F : M N is bijective and both F and F 1 are di ff erentiable, then F is called a di ff eomorphism ....
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 Spring '10
 Wong
 Geometry, Vectors

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