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In order to explain to non-experts what is a vectorfield, one usually describes an assignemnt of an arrow to each point of space, and this works quite well, also when moving to manifolds (where a generalized arrow will be a tangent vector).

My question is: What are similar objects that can help imagining differential forms?

Positive qualities for such object would be (for example) that it helps justifying easily change of coordinate formulas and formulas for pullbacks via functions, or that it "easily drawable", or that it helps understanding more complicated differential-form-based concepts (e.g. connections, cohomology groups, etc.).

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n-form defines a volume element and orientation on a n-dimensional manifold. –  Andrey Rekalo Jun 3 '10 at 18:01
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@Steve Huntsman: Thank you for the link! –  Andrey Rekalo Jun 3 '10 at 18:08
    
Perhaps this will help: mathoverflow.net/questions/4648/when-to-pick-a-basis/4900#4900 –  Ilya Grigoriev Jun 3 '10 at 22:36
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9 Answers 9

If you look at Misner-Thorn-Wheeler's book, Gravitation, you will see heroic attempts to draw pictures of differential forms. But I think this is misguided. Not everything can be drawn directly as a picture.

To me, a 1-form is a measuring instrument defining a unit of speed for a vector field. Without a $1$-form, there is no natural way to measure the length of a vector field or the speed of its integral curves. A 1-form is the simplest co-ordinate-free way to do so.

Higher exterior powers of vectors and forms have corresponding but more elaborate explanations.

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very interesting explanation.. so a 2-form is [something giving] a coordinate-free way to measure the speed of "diffusion of particles" along a [given] 2-plane distribution, right? –  Mircea Jun 4 '10 at 19:04
    
Well, the 2-dimensional analogue of a vector field is a field of infinitesimal parallelograms. So a 2-form is an instrument for measuring the area of such an infinitesimal parallelogram. –  Deane Yang Jun 4 '10 at 22:22
    
A 1-form certainly assigns a number to a vector, but to me it seems kind of misleading to think of this number as the "speed" of the vector because "speed" shouldn't be linear in the vector. Sure, the speed should scale linearly with the vector. But it shouldn't be additive the way a 1-form is. Two vectors pointing in opposite directions should have the same speed, not the opposite speed. A "speed" should not have a codimension-1 hyperplane of vectors in its kernel. And so on. –  Tim Campion May 8 '13 at 2:21
    
Moreover, the most important subclass of 1-forms are the closed ones, whose integrals along curves are locally path-independent. If you had a notion of "speed" with this property, then its integral should be a distance. Then a "closed speed" gives you distances which are locally path-independent. Very weird. –  Tim Campion May 8 '13 at 2:22
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Tim, you make some good points. I shouldn't have said just "speed". It's a "directional speed" but in the following sense: Imagine a bunch of evenly spaced ordered set of parallel planes in space and an object traveling through the planes. The orientation of the planes represents the "direction" in an affine invariant way (i.e., without using the notion of angle or the inner product). Then you can measure how quickly the object passes through the parallel planes. And since the parallel planes are ordered, you can also assign a sign to this speed. –  Deane Yang May 8 '13 at 3:36
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$p$-forms are what you can integrate over submanifolds of dimension $p$. While this may sound way too operative to be called an intuition, it'll get you quite far.

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The $k$-forms that are easiest to describe are those with $k \in \{0,1,n-1,n\}$. A 0-form on an $n$-manifold is a function. A 1-form on an $n$-manifold, if you imagine it in $n+1$ dimensions, is like an arrangement of shingles on a roof: At each point of the manifold, it defines a directional slope, which as other people have said, is the same as a dual vector on tangent vectors. An $n$-form is a density, i.e., an entity that you can integrate over the manifold. And an $(n-1)$-form is a flux (like, say, describing oil coming out of a well): At each point it has a null tangent direction, and it assigns a non-zero volume to each cross section.

Of course you can think of any $k$-form as a $k$-dimensional flux, and for general values of $k$ you might as well. But when $k$ is 1 or $n-1$, it is somewhat easier to visualize the condition that the form is closed. A 1-form is closed when the shingles locally mesh as the slope of a smooth roof, i.e., the form is locally integrable. An $(n-1)$-form is closed when the flux is locally conservative, which for instance is the case with fluid flow. In fact, theorem: A closed, non-zero $(n-1)$-form is equivalent to a 1-dimensional foliation with a transverse volume structure.

The reason that other values of $k$ are harder is that while you do get an entirely analogous algebraic integrability condition when the form is closed, you might not get the same kind of geometric integrability. A non-zero 1-form has an $(n-1)$-dimensional kernel at each point. (Although the visualization that I suggested is in $n+1$ dimensions, it is also true in $n$ dimensions that these tangent hyperplanes mesh when the 1-form is closed.) A non-zero $(n-1)$-form has a 1-dimensional kernel at each point. But a $k$-form for other values of $k$ doesn't usually have a kernel. (Okay, a maximum rank 2-form in odd dimensions also has a 1-dimensional kernel, and it is equivalent to a 1-foliation with a transverse symplectic structure.)

I have heard the statement that only 1-forms and 2-forms are any good. (Well, that's an overstatement, but they are more important than the others except for maybe $0$ and $n$.) In particular, symplectic forms show up a lot, so it is important to try to imagine them even though by definition they have no kernels. I think of a symplectic form as a calibration for a local complex structure. (Or an almost complex structure, which might be all that exists globally.) I.e., among the different tangent 2-planes of a symplectic $2n$-manifold, the ones that are complex lines have the greatest pairing with the symplectic form, while the ones that are real planes have vanishing pairing, and the pairing minimum is achieved by complex lines with the wrong orientation.


One more remark: The geometric picture of a foliation with a transverse volume structure holds for closed $k$-forms that are also non-zero simple forms (i.e., wedge products of linearly independent 1-forms). I think it's a theorem that any closed $k$-form is locally a sum of closed, simple $k$-forms. If that's correct, then that's also a way to visualize a closed $k$-form, as an algebraic superposition of volumed foliations. $k=1$ and $k=n-1$ are special cases in which every non-zero form is simple.

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I tend to interpret the 'goodness' of the $1$- and $2$-forms as saying that we do not understand $k$-forms for $k\geq3$ well enough :P –  Mariano Suárez-Alvarez Jun 4 '10 at 4:59
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Re "any good" remark: 2-forms are curvature forms of a connection, so obviously they are very important; their exterior powers represent higher Chern classes, so they are important as well. 3-forms also arise naturally, for example, there is a canonical invariant 3-form on any compact Lie group, given by $(X,Y,Z)\mapsto K([X,Y],Z)$ on the level of the Lie algebra. J.-L. Brylinski wrote a whole book treating integral 3-forms (3-curvature forms of gerbes). –  Victor Protsak Jun 4 '10 at 15:24
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A one-form assigns to each vector tangent to a manifold a real number in a linear way. You may think of a vector tangent to a manifold as being determined by two points on the manifold that are "infinitesimally close", and hence view a 1-form as a function from such infinitesimal pairs of points to the real numbers. This analogy persists in higher dimensions; an n-form is a way of assigning n-points which are mutually infinitesimally close a real number (with the additional assumption that this assignment be antisymmetric). This is more than an analogy. The techniques of synthetic differential geometry allow for a rigorous definition of n-forms this way and it can be proven to line up with the classical definition. The details can be found here:

http://home.imf.au.dk/kock/van00.PDF

Anders Kock has many of his papers on his webpage including some which explain the concept of connections in this language. The advantage of the synthetic approach is that the definitions seem to line up with heuristic arguments often used to think about these objects. Here is the webpage:

http://home.imf.au.dk/kock/

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The definition of Koch speaks of "assigning a number to infinitesimal simplexes, so that the number is zero if the simplex has two coincident vertices".. this sonds like a nice definition to use for cohomology! –  Mircea Jun 4 '10 at 19:40
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One-forms are coordinates on the tangent space at a given point. Given a basis of a n-dimensional vector space, the dual basis is just the nth coordinates functions on the vector space in the given basis. Of course, if the basis of vector depends on the point, its dual basis also changes as we move the point on the manifold.

n-forms in n-dimensional space are a way to compute volume, or more generally the extensive quantities that appears in physics: mass, electric charge etc. The "value" of the n-form at a point is the infinitesimal mass located at the point.

Explaining forms of intermediate degree is a bit tricky. I think that some physical intuition can help again. Given a thin surface of non-homogeneous material in 3-dimensional ambient space, we want to describe how the mass is distributed on the surface. At each point of the surface, we want a 2-dimensional element of "volume". This infinitesimal mass depends on the point chosen on the surface. The distribution of 2-dimensional mass can be described by a 2-form. That form retains two informations: the way the tangent plane to the surface is oriented in space (which is completely determined by the normal vector to the surface), and the infinitesimal mass value.

Arguably, such an explanation suggests that a p-form can be "integrated" so as to give some p-dimensional manifold, which is completely wrong. Still it provides some intuition for p-forms, which are meant to be integrated on manifolds, and it can be a starting point for a more elaborate answer.

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A 2-form in a 3-manifold does not retain the information of the orientation of the tangent plane to the surface. It rather provides a mass density (or charge density, to account for the sign) for each surface. –  Marcos Cossarini Jul 7 '10 at 3:37
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90% of the time, all you need to know about k-forms is that they are something that turns a k-tuple of vector fields into a function on your manifold. (Aka they eat vector fields and give functions.)

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47.5% of all statistics is made up on the spot! (FTR: I agree with the other 95% of your answer) –  Victor Protsak Jun 4 '10 at 4:09
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I'm sure Mircea you've already thought of what I'm about to say, but if you cheat a little (viz. you assume you're manifold has a metric) you can always think of cotangent vectors as tangent vectors which act on others by scalar product.

I guess as one imagines the wedging of tangent vectors as (higher dimensional) parallelograms one can apply a similar trick.

p.s. Another idea might be to view cotangent vectors as their kernels, so hyperplanes in tangent space. This gives a picture of the projectivisation of the cotangent space. Maybe this trick can be extended as well to differential forms?

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I agree about 1-forms and 1-vectors, but I'm not sure of the dualtity between k-forms and k-vectors, for k different from 0,1,n-1,n (if n is the dimension of the manifold)... my intuition would then be that the dual of k-forms are just "simple k-vector"-fields. To justify that, observe that a form should be something that integrates on a submanifold, and the vector $e_1\wedge e_2+e_3\wedge e_4$ in $\mathbb R^4$ is not representing the tangent of a submanifold. –  Mircea Jun 4 '10 at 18:51
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Terry Tao has wonderful article in the PCM on differential forms. I frequently refer to it when I'm trying to get my head around some concept.

http://terrytao.wordpress.com/2007/12/25/pcm-article-differential-forms/

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I personally learned a lot from trying to formulate and visualize the analogue of differential geometry over $GF(2)$, also known as differential logic. There is an exposition here.

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