9c6c8fc5bd-p-76.pdf

# 9c6c8fc5bd-p-76.pdf - 7.5 ISOMETRIAS 139 entonces que X(t...

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7.5. ISOMETR ´ IAS 139 entonces que X γ ( t ) , γ ( t ) T γ ( t ) Q , y tiene sentido considerar la apli- caci´on diferenciable [ a, b ] R t g ( X γ ( t ) , γ ( t )) . Definici´on 7.4.1 Definimos la circulaci´on de un campo X X ( Q ) a lo largo de una curva diferenciable γ en Q como la integral b a g ( X γ ( t ) , γ ( t )) dt. (7.2) Obs´ ervese que esta definici´on coincide con la circulaci´on de X a lo largo de γ , esto es: b a X γ ( t ) ( γ ( t )) dt. En particular, resulta independiente de la reparametrizaci´on de γ (Pro- posici´on 6.5.1). Por otra parte, de las propiedades de la circulaci´on de formas cerradas y exactas (Secci´on 6.5) resultan inmediatas las propiedades an´alogas para la circulaci´on de campos irrotacionales y conservativos. As´ ı, si X es conservativo entonces la circulaci´ on de X a lo largo de una curva depende s´olo de los extremos de la curva . Concretamente, la circulaci´on de X = grad f a lo largo de la curva γ verifica b a g ( X, γ ) dt = f ( γ ( b )) - f ( γ ( a )) .

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