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Unformatted text preview: 6.1. AFFINE CONNECTION 145 Then, if X = X i e i is a vector field, then X = X i i . If Y = Y j e j is another vector field then X Y = X i n e i ( Y k ) + k ji Y j o e k = nh dY k + k ji Y j i i ( X ) o e k . That is, X Y = X i ( i Y ) k e k where ( i Y ) k = e i ( Y k ) + k ji Y j We will often write simply i Y k meaning ( e i Y ) k . This should not be con fused with the covariant derivative of the scalar functions Y k . The tensor field Y of type (1 , 1) with components i Y k , that is, Y = i i Y = i Y k i e k . is called the covariant derivative of the vector field Y . The components of the covariant derivative are also denoted by Y k ; i , in con trast to partial derivatives i Y k , which are also denoted by Y k , i . In the coordinate frame e i = i the covariant derivative takes the form i Y k = i Y k + k ji Y j ....
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.
 Spring '10
 Wong
 Geometry

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