To see a tensor

  • #1
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A vector is drawn as an arrow, a covector (one-form) as a series of parallel lines. Is there a way to pictorially represent a tensor of rank greater than one? I want to have an intuitive/geometric sense of what it means to parallel transport such an object, and without a picture I don’t have one.
 

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  • #2
The outer product of two vectors is a simple tensor.

[tex]u^a v^b = w^{ab}[/tex]

So for visualization purposes, you can imagine a tensor as a pair of arrows emanating from the same point.
 
  • #3
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The book Gravitation, by Misner, Thorne, Wheeler, discusses this ad nauseam. I recommend you take a look at that.
 
  • #4
tiny-tim
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Mtw

Yeah … MTW really rocks on this! :smile:
 
  • #5
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Thanks; I just happen to have that massive book on loan from the UNH physics library right now. I like Phlogistonian's idea, too. I guess if a (2,0) tensor can be imagined as a pair of arrows emanating from the same point then a (0,2) tensor like the metric tensor can be visualized as two overlapping sets of parallel lines that curve along with the coordinate system, although I suspect that there are limitations to such things...
 
  • #6
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a covector (one-form) as a series of parallel lines.
I've never known much about visualizing tensors. Could someone explain how this visualization works? It's obvious that a vector can be thought of as an arrow, but I'm not sure how a covector would be a series of parallel lines to be honest.

One visualization I am familiar with applies only to differential n-forms. It grows out of integration theory. Basically, you think of an n-form as being the sort of "density" or volume element that you would integrate over a manifold.
 
  • #8
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I've never known much about visualizing tensors. Could someone explain how this visualization works? It's obvious that a vector can be thought of as an arrow, but I'm not sure how a covector would be a series of parallel lines to be honest.

One visualization I am familiar with applies only to differential n-forms. It grows out of integration theory. Basically, you think of an n-form as being the sort of "density" or volume element that you would integrate over a manifold.
Well, not really parallel lines, but parallel surfaces. Think of the function [tex]f(x)[/tex]. The gradient [tex] df[/tex] or [tex]\nabla f[/tex] defines a one-form, and if you contract with a vector [tex]\vec{v}[/tex], you get the directional derivative of [tex]f[/tex] in the direction pointed by [tex]\vec{v}[/tex].

If you take a curve with tangent vector [tex]\vec{v}(\lambda)[/tex] and you integrate [tex]\langle df, \vec{v} \rangle[/tex] along the curve, then by the fundamental theorem of calculus, you are integrating [tex]df/d\lambda[/tex], or how much f changes. Now think of surfaces [tex]f(x) = const[/tex], where each surface is evaluated for a different constant, and the constants are say, 1 unit apart. The integral you just computed tells you how many surfaces you have to cross as you move along the curve. MTW calls this "bongs of a bell" but anyway. So people visualize covectors, oneforms of the form [tex]df[/tex], as stacked surfaces, like layers of an onion.
 
  • #9
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Very nice. That's a neat way of doing it. But am I right in supposing it only works for exact forms (since you visualize it as the level surfaces of a function f such that the one-form is given by df)?
 
  • #10
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Yup.
 
  • #11
robphy
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Very nice. That's a neat way of doing it. But am I right in supposing it only works for exact forms (since you visualize it as the level surfaces of a function f such that the one-form is given by df)?
It can always work locally.
 
  • #12
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It can always work locally.
If said 1-form is not closed...? What is [tex]\omega = y\, dx[/tex] the differential of?
 
  • #13
Hurkyl
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If said 1-form is not closed...? What is [tex]\omega = y\, dx[/tex] the differential of?
I think he means pointwise (e.g. draw the pictures in the tangent space at the point of interest), rather than being perfectly accurate within an entire open neighborhood.
 
  • #15
robphy
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Very nice!

Do you have a pdf where the individual pages are separated?
Thanks.
Sorry... I don't have that with letter-size pages.
...but here is an early version:
http://physics.syr.edu/~salgado/papers/VisualTensorCalculus-AAPT-01Sum.pdf [Broken]
 
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