differential geometry w notes from teacher_Part_37

differential geometry w notes from teacher_Part_37 - 73 3.2...

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3.2. ORIENTATION AND THE VOLUME FORM 73 3.2.5 Pseudotensors and Tensor Densities Let E be an oriented vector space and B be the set of all bases on E . Then the orientation is a function o : B → R defined by o ( e i ) = + 1 if e i is positively oriented - 1 if e i is negatively oriented A pseudo-tensor T on a vector space E assigns, for each orientation o of E a tensor T o such that when the orientation is reversed the tensor changes sign, i.e. T - o = - T o . That is, a pseudo-tensor is a collection of two tensors T + and T - , one for each orientation. A pseudo-tensor field on a manifold is a smooth assignment of a pseudo- tensor to each point of the manifold. Let x i α = x i α ( x β ) be a local di eomorphism and J ( x β ) = det x i α x j β . Since this transformation is a di eomorphism J , 0. Thus, there are two cases J > 0 and J < 0. A
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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.

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differential geometry w notes from teacher_Part_37 - 73 3.2...

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