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- Thread starter snoopies622
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[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.

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tiny-tim

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Yeah … MTW really rocks on this!

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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.

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"Visualizing Tensors"

at

www.opensourcephysics.org/CPC/abstracts_contributed.html

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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.

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Yup.

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It can always work locally.

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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?

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Hurkyl

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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.If said 1-form is not closed...? What is [tex]\omega = y\, dx[/tex] the differential of?

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Very nice!

"Visualizing Tensors"

at

www.opensourcephysics.org/CPC/abstracts_contributed.html

Do you have a pdf where the individual pages are separated?

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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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