differential geometry w notes from teacher_Part_37

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online