I am looking at this document I do not understand how the author gets 5.12 and 5.13 on page 133. I think the matrix of partials should be the transpose of the one shown. Even so I still can't figure out how you get 5.13. Any help would be appreciated.
I am trying to improve my understanding of Lie groups and the operations of left multiplication and pushforward.
I have been looking at these notes:
I am trying to figure how one arrives at the following:
dxμ∂ν = ∂xμ/∂xν = δμν
dxμ is the gradient of the coordinate functions = basis of cotangent space
∂ν = basis of tangent space
I know that dual vectors 'eat' vectors to produce scalars. Is this demonstrated by absorbing d into ∂...
OK, I have modified my original questions. I think (hope) I now have a better understanding. Perhaps somebody could critique this. However, I am still a little confused about the relationship between λ and Ω and the difference in exponents. Thanks.
A manifold, M, is...
I am confused about conformal transformations on Riemannian manifolds. Here's what I have so far.
1. Under a conformal transformation the metric changes by:
g' -> Ω2g
2. Under a Weyl transformation the metric changes by:
g' -> exp(-2f)g
3. Any 2D Riemann manifold is locally conformally...
dVμ = (∂Vμ/∂xη)dxη where Vμ is a contravariant vector field
I believe the () term on the RHS is a covariant tensor. Is the dot product of () and dxη a scalar and how do I write this is compact form. I know how this works for scalars but am not clear when tensors are involved.