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3-vector, 2-vector?

by fisico30
Tags: 2vector, 3vector
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fisico30
#1
Aug26-08, 04:24 PM
P: 374
I wonder if anyone has ever heard this terminology. What is the meaning of a 3-vector?
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Varnick
#2
Aug26-08, 04:32 PM
P: 77
The number just refers to the number of dimensional components. A 2-vector has components in two dimensions (X and Y) whereas a 3-vector has components in three dimensions (X Y and Z).

V
Hurkyl
#3
Aug26-08, 04:36 PM
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Sometimes, it refers to wedge products of vectors -- i.e. a 2-vector would have a planar direction and magnitude, while a 3-vector would have a spatial direction and magnitude.

fisico30
#4
Aug27-08, 04:38 PM
P: 374
3-vector, 2-vector?

i knew I was not asking it right! Sorry. I actually meant 0-form, 1-form, 2-form...
I believe they are referred to the idea of manifold, but I am not actually sure I know what a manifold is. Is it just any entity that resembles a plane if zooming at a point? That does not make sense right... any example?
Varnick
#5
Aug28-08, 03:03 AM
P: 77
A manifold is a non-orthogonal co-ordinate system, where euclidean geometry applies at a local level. Consider the surface of the earth. At a large scale, Euclidean geometry does not apply, but if you zoom in to a relatively smaller scale, Euclid's axioms begin to apply again. One can also think of it as a patchwork, lots of small Euclidean systems joined together to form a non-Euclidean manifold.

V
fisico30
#6
Sep2-08, 11:58 AM
P: 374
Clear answer Varnik. So you say it is a coordinate system, which is a way to describe/ locate something in 3D space.
The spherical, rectangular, cylindrical coord systems can then also be a called manifolds if they are locally very similar to a flat plane (2D cartesian system?).

the patches you talk about are 2D (x and y). Can the manifold be locally similar to a 3D Cartesian system? Any example.

Can manifolds be though as something else too, besides coord. systems?
Here a reply I found on DR. Math website. Hope it helps(me and others):

"Take a part of a plane (which is two-dimensional, right?). Cut out a
part. Now, this plane is actually made of rubber. So you can pull it
and stretch it and squish it and curve it and do most anything to it.
(but you pop it if you make a sharp point or edge, so you can't do
that - a type of mathematician called a 'topologist' loves to do
this.) Now, you can twist it around some and get part of a sphere,
right? if you put this sphere in three-dimensional space, that means
you have a 2-manifold in 3-space. 3-space just means three-
dimensional space. (mathematicians like to sound cool by saying
3-space.) Similarly, take a long line made of rubber (very thin rubber!). If you
stretch it and curl it and put it in a plane, we call it a 1-manifold
in 2-space. If we put it in a three-dimensional space, we call it a
1-manifold in 3-space.

Now for the really mind-boggling part. Take a portion of three space
(your room, for example), and twist it around and stretch it. Put it
in four-dimensional space. That's what's called a 3-manifold in
4-space. There's no real way to picture this, which is why
mathematicians tend to rely on equations, not just on pictures!"

It seems that ANYTHING that has is described by n-dimensions(n degrees of freedom) can be a n-manifold. It could be an object, a field, etccc.
Varnick, Do you agree or am I completely off?

Thanks!
Varnick
#7
Sep2-08, 12:42 PM
P: 77
You seem to have the idea, I cannot give an easy example of a 3-manifold, although spacetime is an obvious 4-manifold, it is hard to picture.

V


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