Differential forms on R^n vs. on manifold

In summary, differential forms over R^n and on manifolds are essentially the same, but on a general manifold, one must use local charts and the local coordinate form can vary from chart to chart. The exterior product and derivative have different properties, but these differences can be explained by using the concept of tangent spaces.
  • #1
Kris-L
2
0
First time looking at differential forms. What is the difference of the forms over R^n and on manifolds? Does the exterior product and derivative have different properties? (Is it possible to exaplain this difference without using the tangent space?)
 
Physics news on Phys.org
  • #2
Kris-L said:
First time looking at differential forms. What is the difference of the forms over R^n and on manifolds?
Can you define "over ##\mathbb{R}^n##" versus "on a manifold"?
Does the exterior product and derivative have different properties?
Depends on what you consider a property.
$$\begin{equation*} d(\omega_1 \wedge \omega_2)= d\omega_1 \wedge \omega_2 + (-1)^k \omega_1 \wedge d\omega_2 \\ \textrm{ for all }\omega_1 \in \wedge^k(U)\, , \,\omega_2 \in \wedge^l(U) \end{equation*}$$
is the rule for multiplications in the exterior (Graßmann) algebra, whereas derivatives obey the product (Leibniz) rule
$$
d(f\cdot g)=df \cdot g + f \cdot dg
$$
This can be considered as a difference, although the comparison doesn't match exactly.
(Is it possible to explain this difference without using the tangent space?)
Not really. You can use a different language: sections, pullbacks, and vector bundles. But in the end, we are talking about tangent spaces.

You might want to read:
https://www.physicsforums.com/threads/why-the-terms-exterior-closed-exact.871875/#post-5474443
https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/#toggle-id-1
 
  • #3
The exterior derivative business is a way to systematize the sorts of derivatives that are already used in Euclidean 3-D vector calculus: the gradient, the divergence, the curl.

Then some of the amazing facts about vector calculus can then be seen as special cases of much more general facts. For example, we know that the curl of the gradient of a scalar is always zero. Similarly, the divergence of the curl of a vector is always zero. In exterior calculus, these are two instances of the more general fact that ##d^2 = 0##. Applying the exterior derivative twice always produces zero.

Then there are two facts relating integrals of different numbers of dimensions:

##\int \vec{A} \cdot \vec{dl} = \int (\nabla \times \vec{A}) \cdot \vec{dS}##
(integrating a vector field around a closed loop produces the same result as integrating the curl of the vector over the surface enclosed by the loop)

##\int \vec{A} \cdot \vec{dS} = \int (\nabla \cdot \vec{A}) dV##
(integrating a vector field over a closed surface produces the same result as integrating the divergence of the vector over the volume enclosed by the surface)

In the exterior calculus, these are two different instances of the more general "Stokes theorem", which applies to objects of any degree.
 
  • Like
Likes Ibix
  • #4
Only global properties of the forms can be different in R^n and in other manifold. Locally defined operations ( like different differentiations, exterior product, i_v etc) have the same properties
 
  • Like
Likes atyy and mathwonk
  • #5
Kris-L said:
First time looking at differential forms. What is the difference of the forms over R^n and on manifolds? Does the exterior product and derivative have different properties? (Is it possible to exaplain this difference without using the tangent space?)
I am not sure what the question is. I would say that the answer is no, they are the same as R^n is a manifold. The only thing that changes are the charts that define our manifold. On R^n there happens to be one global chart that is the identity map so that makes things easier.

For a general manifold only patches of it are diffeomorphic to R^n so you have charts for these different patches. When doing computations with forms one often needs to work with local coordinates which involves mapping patches of manifold to R^n and doing the computations in the exact same fashion a in R^n.

Different patches have different chart maps so that the form can end up being a different function in different charts. So the main differences are this. On a general manifold, you have to do computations in multiple local charts and the local coordinate form can vary form chart to chart. This isn't an issue in R^n.
 
  • Like
Likes atyy

1. What is the difference between differential forms on R^n and on a manifold?

The main difference between differential forms on R^n and on a manifold is the dimensionality. Differential forms on R^n are defined in Euclidean space of n dimensions, while on a manifold, they are defined on a curved space that may have a different dimensionality. Additionally, differential forms on a manifold take into account the local coordinate system, while on R^n, they are independent of any specific coordinate system.

2. How are differential forms on R^n and on a manifold related?

Differential forms on R^n and on a manifold are related in that the latter can be thought of as a generalization of the former. In other words, differential forms on R^n can be seen as a special case of differential forms on a manifold when the manifold is flat and has the same dimensionality as R^n.

3. What is the significance of differential forms on R^n and on a manifold?

Differential forms on R^n and on a manifold are important mathematical tools used in many areas of physics and engineering. They provide a way to describe and manipulate quantities, such as vectors, tensors, and differential equations, in a coordinate-independent manner. This makes them particularly useful in fields where the underlying space is curved or non-Euclidean.

4. How are differential forms on R^n and on a manifold used in practical applications?

Differential forms on R^n and on a manifold are used in a variety of practical applications, such as in the study of fluid mechanics, electromagnetism, and general relativity. They are also used in computer graphics and computer vision to represent and manipulate geometric objects. In addition, differential forms play a crucial role in the development of numerical methods for solving differential equations.

5. Are there any limitations to using differential forms on R^n and on a manifold?

One limitation of using differential forms on R^n and on a manifold is that they can be quite abstract and difficult to visualize, especially in higher dimensions. Additionally, the calculations involved in manipulating differential forms can be complex and time-consuming. However, with the help of modern mathematical software, these limitations can be overcome, making differential forms a powerful tool in many fields of study.

Similar threads

  • Differential Geometry
Replies
10
Views
708
  • Differential Geometry
Replies
7
Views
2K
  • Differential Geometry
Replies
4
Views
2K
  • Differential Geometry
Replies
21
Views
3K
Replies
13
Views
526
Replies
6
Views
343
  • Differential Geometry
Replies
2
Views
584
  • Differential Geometry
2
Replies
55
Views
7K
  • Differential Geometry
Replies
13
Views
996
  • Differential Geometry
Replies
5
Views
3K
Back
Top