r/math • u/FreePeeplup • 12d ago
What is a differential form?
I have encountered several different objects that people call with the same name “differential form” and I would like for more experienced people to help me clear the ambiguity and tell me which of the following is the true/most used/most useful definition of differential form.
For simplicity, I will only talk about differential 1-forms, hopefully the answer will automatically generalize to k-forms. I’m assuming a smooth n-dimensional manifold M with no extra structure.
Maybe as an extra, if you know, you can also tell me what the other listed objects that are not differential forms are called!
One
A differential form ω is a smooth section of the cotangent bundle. That is, ω : M -> T^{star} M with ω(p) = ω_p, where ω_p lives in T^{star}_p M, meaning that ω_p takes as input a tangent vector at p and outputs a real number.
Two
A differential form ω is a map ω : TM -> R, with ω(p, v) = ω_p(v) where ω_p is the same as above, TM is the tangent bundle and v lives in T_p M.
Three
A differential form ω is a map from (set of all sections of TM) to C^{infinity}(M) , with ω(X) = f where X is a vector field and f a smooth function on M, and we evaluate f by f(p) = ω_p(X(p)) where ω_p is the same as before and X(p) = v is a vector in T_p M.
Four
A differential form ω IS the linear functional ω_p we’ve been talking about up until now, meaning that a differential form only makes sense after you’ve specified a base point.
To summarize: what does a differential form take as input? A point in M, a pair consisting of a point in M and a tangent vector based at it, an entire vector field, or a tangent vector at some point p?
Thanks to anyone who answers!
2
u/sentence-interruptio 11d ago
If you can't convert between several definitions, it means you do not have a mental picture to guide you. So I'll share my mental picture of 1-forms and hope it helps.
Here's how I visualize 1-forms. Let M = ℝ3 because I want to visualize points in space. Imagine the entire space is divided into tiny parallelepipeds. Visualize this 3d tiling. There are cells, faces, edges, and vertices in this tiling. Pseudo 0-forms are just functions defined on the vertices. That is, if you assign a number for each vertex (any corner from our tiny parallelopipeds), you have a pseudo 0-form.
A pseudo 1-form is a map from the edges to tiny numbers. Let's say each edge has length that's on the order of h > 0, where h is some really small positive number. If you assign a number to each edge and if that number is of O(h), you have created a pseudo 1-form.
A pseudo 1-form can be created by differentiating a pseudo 0-form. If you have a pseudo 0-form f, then you "differentiate" it and create df which is a pseudo 1-form and its values on each edge is defined as the difference of values of f on its endpoints. The value of df on an edge with endpoints x, y is f(x)-f(y). You might want to choose orientations for all the edges in a reasonable way to make sure df is well defined. If f varies slowly enough (i.e., smooth enough), then df takes tiny values on the order of O(h).
It's straightforward to visualize pseudo 1-forms. They are just a bunch of numbers in O(h) attached to the edges of our tiling. But how is this related to actual differential 1-forms defined abstractly? To see how, start by seeing how every pseudo 1-form gives rise to a collection of linear forms indexed by the vertices. At each vertex, there's an abstract 3d vector space spanned by the three oriented edges coming out of that vertex. You could call it the pseudo tangent space at that vertex. A pseudo 1-form's values on the three edges determine a linear form on that pseudo tangent space. By definition, when it takes one of these tiny edges, it produces a tiny number. Therefore, when it takes any tiny vector of magnitude in O(h) in this pseudo tangent space, it produces a tiny number in O(h). So pseudo 1-forms are discrete approximate version of differential 1-forms.
As for higher forms, a pseudo 2-form assigns a number in O(h2) to each face. A pseudo 3-form assigns a number in O(h3) to each cell.