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In this chapter we will introduce and discuss at length one of the ways that physicists sometimes visualize “nice” differential forms. In essence, we will be considering ways of visualizing one-forms, two-forms, and three-forms on a vector space. That is, we will find a “cartoon picture” of $ \alpha_p \in T^*_P R^2 $ and $ \alpha_p \in T^*_P R^3 $ . Our picture of $\alpha_p$ will be superimposed on the vector space $ T^*_P R^3 $ . This perspective is developed extensively in Misner, Thorne, and Wheeler’s giant book Gravitation. In fact, they make some efforts to develop the four-dimensional picture (for space-time) as well, however we will primarily stick to the two and three-dimensional cases here. Section one focuses on the two-dimensional case and sections two through four focuses on the three-dimensional case. Then in section five we expand our cartoon picture to general two and three-dimensional manifolds. Again, this cartoon picture really only applies to “nice” differential forms, of the kind physicists are more likely to encounter, but it is still useful for forms and manifolds that are not overly complicated. Finally, in section six we introduce the Hodge star operator. Our visualizations of forms in three dimensions provide a nice way to visualize what the Hodge star operator does, which makes this a nice place to introduce it. Despite the power and usefulness of the way of visualizing differential forms in physics developed in this chapter, it is rarely encountered in mathematics. One of the reasons for this is that in reality it is not a completely general way of considering forms; when dealing with dimensions greater than four or with more abstract manifolds or with forms that are not “nice” in some sense it breaks down.

from A Visual Introduction to Differential Forms and Calculus on Manifolds (Visualizing One-,Two-, and Three forms Chapter $5$)

One of the Example Techniques

One of the sample tricks he mentions is that we can imagine number of lines the vector pierces as the "magnitude" of the vectors component.

Where $dy ([a,b]^T) $ refers to the $y-$component of the vector $[a,b]^T$ which is $b$.

enter image description here

But later says:

A necessary, but not sufficient, requirement is that the distribution given by the kernel of the differential form be integrable. This would give what is called a foliation of the manifold which would allow the line-stacks, plane-stacks, or tube-bundles to line up. Exploring and explaining this is beyond the scope of this book. It is simply sufficient to realize that even a one-form on $R^3$ as simple as $xdy + dz$ can not be visualized using method two.

enter image description here

Question

Why does the physicist trick of visualizing differential forms breakdown in dimensions $> 4$?

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  • $\begingroup$ Have you tried doing it in 4+ dimensions? $\endgroup$ Jan 16 '21 at 22:57
  • $\begingroup$ @AndersSandberg I'm actually kind of new to this whole visualizing business but I can't intuitively see what would go wrong $\endgroup$ Jan 17 '21 at 5:07
  • $\begingroup$ @NiharKarve Yes believe so I was copying the book exactly as is $\endgroup$ Jan 17 '21 at 11:06
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    $\begingroup$ OK, I understand and it makes sense, but (my 2 cents) it will be more difficult for you to get people to answer if they need to get a copy of this book and read the chapter as background reading for your question. At least, for me, that is a roadblock. $\endgroup$
    – Andrew
    Apr 2 '21 at 8:39
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    $\begingroup$ That is extremely useful, thanks for doing this @MoreAnonymous. I don't have time right away to write a full answer, but if no one comes back I can probably write one in the next few days. A short version would be that you probably know that some vector fields can be written as a gradient of a scalar field, in which case the level sets of the scalar field are useful visualization tools, but not every vector field can be written as the gradient of a scalar. An analogous set of statements is also true for forms, and that is what the author is saying. However this isn't specific to 4 dimensions. $\endgroup$
    – Andrew
    Apr 2 '21 at 12:53
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The author is probably referring to the fact that in two and three dimensions, all bivectors are simple (i.e., that can be expressed as the exterior product of two vectors)... and so can be visualized as an oriented planar tile. (One can say a similar thing for two-forms.)

https://en.wikipedia.org/wiki/Bivector

In general, bivectors would be the sum of "simple bivectors"... and presumably requires a set of planar tiles [at each point in a bivector field]. A physics example would be the electromagnetic field tensor $F_{ab}$ as a generally non-simple two-form.

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