Differentiable Manifolds

1. Jul 28, 2015

mikeeey

Hello every one

Can one say , that
A globle coordinate chart is a cartesian coordinate
And a local coordinate chart is any kind of curvilinear coordinate ?

Thanks

2. Jul 28, 2015

Fredrik

Staff Emeritus
No. A global chart doesn't have to have orthogonal axes, it just has to be defined on the entire manifold rather than on some proper subset of it.

3. Jul 28, 2015

mikeeey

Then the global coordinate chart would be a rectilinear coordinate ( generalization of cartesian coordinate ) may be affine ( not orthogonal )

4. Jul 28, 2015

mikeeey

5. Jul 29, 2015

WWGD

What do you mean by Cartesian coordinates and axes? What are curvilinear coordinates?

6. Jul 29, 2015

mikeeey

Cartesian coordinates (i.e. The position vector is the same every where in the space ) but curvilinear coordinates ( the coordinates can change in angles from point to point in the space ) i.e the axes are curved

7. Jul 29, 2015

WWGD

Seems you need a map that preserves the inner-product globally, to map Cartesian coordinates to Cartesian coordinates, since orthogonality is/can be defined in terms of the inner -product..

8. Jul 29, 2015

mikeeey

The map is linear and the linear operator is the derivative , sometime linear map ( transformation ) is called the jacobian matrix
Hint : dot product does not mean there is an orthogonality !! Generally non orthogonal but specially orthogonal

9. Jul 29, 2015

WWGD

You're right, but I was referring to the inner-product, not the dot product . But the Jacobian is a local linear map describing the (local) change in the function. The local properties of a fuction and of its Jacobian do not always preserve properties globally. I guess the inner-product is a tensor, so we could see the effect of induced map on tensors.

10. Jul 29, 2015

mikeeey

First inner product is the generalization of dot product ( in tensor algebra mostly u have 4 products of vector spaces 1- tensor product ( outer ) 2- inner product 3- wedge product 4- symmetric product , with these product u can decompose any tensor )
Secondly jacobian matrix is for globle and local coordinates scine its inversible matrix

11. Jul 29, 2015

WWGD

I am not sure I am understanding you. Which map is linear, the derivative? Yes , of course it is, but unless the original map is itself linear, the Jacobian at one point does not globally describe the map.

Jacobian is not always invertible. Take any map between manifolds of different dimension. The Jacobian will be (represented by) an $n \times m$ matrix, which cannot be invertible.. And, as a generalization of the derivative, the Jacobian is a local map describing local change in function, unless map is linear. In what sense is the inner-product a generalization. You mean that in 1D there is an inner -product? Still, it is a covariant 2 -tensor , so we can study the effects of the map using the effects of the induced map between the tangent spaces.

Last edited: Jul 29, 2015
12. Jul 29, 2015

mikeeey

M
my friend if a map is not invertible then the manifold is not homeomorphic ( it must be $n \times n$ so u can use the transition maps ) ( jacobian matrix is used for the same manifold for different charts of the same manifold ) and the inner product is used in higher dimensions ( see wikipedia , inner prod. Is the generalization of dot product )

13. Jul 29, 2015

mikeeey

B
A manifold contains two types of general sapces are the topological space for continuity ...... And other properties and a vector space so we can define tanget spaces and fields and covariant derivatives ... And other properties

14. Jul 29, 2015

WWGD

O.K, maybe I did not understand the maps you are referring to. Yes, chart maps are invertible and actually diffeomorphisms, so, within a chart, what you say is true, between an open set in the manifold and an open subset of $\mathbb R^n$. If your manifold is globally diffeomorphic to some other one, then the Jacobian is an invertible map.

15. Jul 29, 2015

WWGD

That is all clear, I guess I am not understanding well what you are referring to; let me leave this discussion between you and Fredrik.

16. Jul 29, 2015

mikeeey

Thanks for the conversation , by the way im an engineer in mechanics this discussion is far a way from mechanics ,but im interested in general relativity that deals with manifolds .
Thanks

17. Jul 29, 2015

WWGD

Sure, no problem, it is always good to exchange ideas and test your knowledge; can always sharpen up understanding. Unfortunately, we seemed to be at cross-hairs, talking about different things; hope next exchange will be more productive.