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Unformatted text preview: j ∂x i ∂ ∂ ˜ x j , and therefore the same vector V is represented by V = n X i =1 v i ∂ ∂x i = n X j =1 ˜ v j ∂ ∂ ˜ x j = n X i =1 n X j =1 v i ∂ ˜ x j ∂x i ∂ ∂ ˜ x j . This means that ˜ v j = n X j =1 v i ∂ ˜ x j ∂x i , so the corresponding local trivializations η and ˜ η are related by ˜ η ◦ η-1 ( x,v ) = ˜ η ( V ) = ( x, ˜ v ) = ( x,τ ( x ) v ) , where τ ( x ) is the GL ( n, R )- valued function ∂ ˜ x j /∂x i , hence receiving that the reqeusted maps are just the Jacobian matrix of the coordinate transition map. 1 2. Was 3. We 2...
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- Spring '09
- Geometry, Manifold, Differential topology, Xi Xj, Hava Shabtai