Fredrik said: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.
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 curvedWWGD said:What do you mean by Cartesian coordinates and axes? What are curvilinear coordinates?
mikeeey said: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
The map is linear and the linear operator is the derivative , sometime linear map ( transformation ) is called the jacobian matrixWWGD said: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..
mikeeey said: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
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 )WWGD said: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.
mikeeey said: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
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 )WWGD said: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.
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 propertiesWWGD said: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.
mikeeey said: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
Thanks for the conversation , by the way I am an engineer in mechanics this discussion is far a way from mechanics ,but I am interested in general relativity that deals with manifolds .WWGD said: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.
A differentiable manifold is a mathematical concept used in the field of differential geometry to describe a space that locally resembles Euclidean space, but may have a more complex global structure. It is a generalization of the concept of a smooth surface in three-dimensional space, where the notion of differentiability can be extended to higher dimensions.
A differentiable manifold is defined as a topological space that is locally homeomorphic to Euclidean space, and is equipped with a differentiable structure that allows for the definition of differentiable functions on the manifold. This structure includes a collection of charts (coordinate systems) that cover the manifold and transition maps between overlapping charts that ensure consistency in how differentiable functions are defined.
Some common examples of differentiable manifolds include the surface of a sphere, which can be described using two overlapping charts, and the surface of a torus, which can be described using three overlapping charts. Other examples include the real line, the complex plane, and higher dimensional spaces such as n-spheres and projective spaces.
Differentiable manifolds play a central role in many areas of mathematics, physics, and engineering. They provide a powerful tool for studying and understanding complex geometric structures, and are used in fields such as differential geometry, topology, and dynamical systems. In physics, differentiable manifolds are used to describe the geometry of space-time in general relativity, and in engineering, they are used in control theory and robotics.
Smooth manifolds are a special type of differentiable manifold where the transition maps between overlapping charts are infinitely differentiable. This extra requirement allows for the definition of smooth functions on the manifold, which have more desirable properties compared to just differentiable functions. However, the two concepts are often used interchangeably in practice, and many results and techniques in differential geometry apply to both types of manifolds.