
#1
Feb2604, 07:41 AM

P: 1,308

Can a vector be described in the most general terms by two points on a manifold, as a line with a starting and ending point in some manifold?




#2
Feb2604, 08:24 AM

PF Gold
P: 2,885

Hmm depends on which vectors. The ones used in Differential Geometry do not live in the manifold but in the tangent bundle. So You specify them by given a supporting point in the manifold and a free vector in the linear space.
Then you have a bunch of theorems about how such vectors can be mapped to curves in the manifold across the supporting point. 



#3
Mar1204, 07:19 AM

Math
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PF Gold
P: 38,902

A vector can only be defined at a single point. To have a notion of "moving" a vector from one point to another, you need some additional structure such as a metric tensor or a "Riemannian connection". 



#4
Mar1404, 09:18 AM

P: 2,955

What is a vector?Hallsofivy wrote "a vector is a derivative"!. If by this you mean that a vector v = (v^{1}, ... , v^{n}) defines a directional derivative operator at a point P in the manifold and vice versa then I agree. This is Cartan's notion of a vector if I recall correctly? It is based on the notion that displacement vectors are in a onetoone correspondence with directional derivative operators. However a vector can also be defined in otherways too. E.g. a vector can be defined as any quantity v whosse components = (v^{1}, ... , v^{n}) transform as the coordinate displacements dx^{a} which is almost the same thing. Or one can define a vector as a linear map of 1forms to scalars which obeys the Leibnitz rule. 



#5
Mar1404, 11:15 AM

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PF Gold
P: 16,101

There's another way of thinking about it (that is 'natural', but logically strange):
An "arrow" is a onedimensional manifold whose endpoints are different yet in the same place. (This is, of course, the odd part; some fancy logic tricks are needed to make this work) Then, a vector is simply the image of an arrow under a smooth map. 



#6
Mar1504, 07:07 AM

Math
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PF Gold
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#7
Mar1504, 08:26 AM

P: 2,955

Many excellant text discuss these tensors. E.g. Tensors, Differential Forms, and Variational Principles, Lovelock and Rund, page 53 Gravitation and Spacetime  Second Edition, Ohanian and Ruffini, page 308 Classical Electrodynanics  Second Edition, J.D. Jackson, page 535 Classical Field Theory, Francis E. Low, page 259 It's important to understand this when looking at something like Jackson since while he does use partial derivatives to define the transformation of components he never uses anything but the Lorentz coordinates and the Lorentz transformation. I.e. transformations from one set of Lorentz coordinates (ct, x, y, z) to another set of Lorentz coordinates (ct', x', y', z). Lorentz coordinates are defined as (ct,x,y,z) in an inertial frame of reference. Things which are Lorentz 4tensors/vectors are not tensors/vectors in general. One has to be careful when reading Jackson. E.g. the components of 4force in Lorentz coordinates appears in his text as dP^u/dT since the affine connection vanishes in Lorentz coordinates. In generalized coordinates the components of 4force are DP^u/DT (T = proper time) which includes the affine connection Lorentz coordinates and Lorentz 4vectors are obviously a very widely used arena. For those interested in the details of this please see the excelant online notes by Thorne and Blanchard at http://www.pma.caltech.edu/Courses/p...p01/0201.2.pdf 


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