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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 ) + ω k ji Y j o e k = nh dY k + ω k ji Y j σ 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 ) + ω 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 = σ i ⊗ ∇ i Y = ∇ i Y k σ 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 + ω k ji Y j ....
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 Spring '10
 Wong
 Geometry, Vector field, ek, Covariant derivative, Affine connection

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