# To see a tensor

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.

## Answers and Replies

The outer product of two vectors is a simple tensor.

$$u^a v^b = w^{ab}$$

So for visualization purposes, you can imagine a tensor as a pair of arrows emanating from the same point.

The book Gravitation, by Misner, Thorne, Wheeler, discusses this ad nauseam. I recommend you take a look at that.

tiny-tim
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Mtw

Yeah … MTW really rocks on this!

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...

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.

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 $$f(x)$$. The gradient $$df$$ or $$\nabla f$$ defines a one-form, and if you contract with a vector $$\vec{v}$$, you get the directional derivative of $$f$$ in the direction pointed by $$\vec{v}$$.

If you take a curve with tangent vector $$\vec{v}(\lambda)$$ and you integrate $$\langle df, \vec{v} \rangle$$ along the curve, then by the fundamental theorem of calculus, you are integrating $$df/d\lambda$$, or how much f changes. Now think of surfaces $$f(x) = const$$, 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 $$df$$, as stacked surfaces, like layers of an onion.

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)?

Yup.

robphy
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Gold Member
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.

It can always work locally.

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

Hurkyl
Staff Emeritus
Gold Member
If said 1-form is not closed...? What is $$\omega = y\, dx$$ 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.

robphy
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Gold Member
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: