Is there a generalized curl operator for dimensions higher than 3?

Click For Summary

Discussion Overview

The discussion explores the possibility of generalizing the curl operator beyond three dimensions, focusing on theoretical frameworks such as tensor calculus and differential forms. Participants express curiosity about the mathematical foundations and implications of such generalizations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant inquires about a direct method to generalize the rotor operator to more than three dimensions, referencing Wikipedia for context.
  • Another participant notes that understanding the generalization requires knowledge of tensor calculus, which involves higher-order tensors beyond vectors and scalars.
  • A different viewpoint suggests that the generalized curl utilizes skew-symmetric (0,k)-rank forms and can be expressed through exterior algebra with differential k-forms for k greater than three.
  • One participant introduces geometric algebra as a framework for extending curl to N dimensions, emphasizing the role of bivectors and the complexity of surface integrals in higher dimensions.
  • Another participant proposes that curl can be defined using an appropriate volume form in k-dimensional planes, while cautioning about the terminology used for planes in higher dimensions.
  • A later reply mentions the existence of a vector cross product in dimensions 3 and 7, suggesting that curl may also be generalizable to 7 dimensions, referencing previous discussions and papers on the topic.

Areas of Agreement / Disagreement

Participants express various viewpoints on the generalization of the curl operator, with no consensus reached. Multiple competing models and interpretations are presented, indicating an unresolved discussion.

Contextual Notes

Participants highlight the need for careful definitions and the complexity of extending concepts like curl and rotation to higher dimensions, particularly regarding the relationship between surfaces and planes.

Jianphys17
Messages
66
Reaction score
2
  • Hi, i now studying vector calculus, and for sheer curiosity i would like know if there exist a direct fashion to generalize the rotor operator, to more than 3 dimensions!
On wiki there exist a voice https://en.wikipedia.org/wiki/Curl_(mathematics)#Generalizations , but I do not know how you could do...:confused:
 
Physics news on Phys.org
In order to understand the generalisation given in wikipedia you will need to learn a certain amount of tensor calculus. The generalisation is dealing not only with vectors and scalars, but also higher order tensors. The reason the three dimensional case is formulated with vectors only is that in three dimensions there is a direct connection between anisymmetric rank two tensors and vectors.
 
From what I understand, the generalized curl, makes it use of skewsymmetric (0,k)-rank (the k-differential form)!
But i since barely know the tensor calculus formalism..., I've read something on differential forms, therefore it can be expressed through the exterior algebra with differential k-form ( k>3), right? :nb)
 
Jianphys17 said:
  • Hi, i now studying vector calculus, and for sheer curiosity i would like know if there exist a direct fashion to generalize the rotor operator, to more than 3 dimensions!
On wiki there exist a voice https://en.wikipedia.org/wiki/Curl_(mathematics)#Generalizations , but I do not know how you could do...:confused:

Funny you should mention that.

http://hi.gher.space/forum/viewtopic.php?f=27&t=2155

Here's N D curl done with geometric algebra, which is similar to differential forms.

Curl is traditionally a pseudovector. This works only in 3D. A pseudovector is the dual of a plane. So one recasts curl in terms of planes, ie bivectors or 2-forms. It's a chunk of work to get used to, but is essential to extending to N D. Curl is naturally expressed as a bivector or tensor.

Now only in 3D does a bivector define a surface. Surfaces are (N-1)D, planes are 2D. Only in 3D is N-1=2. So all those theorems that involve surface integrals of curl become more involved. To proceed I suspect it is necessary to go back to the roots of EM in special relativity.
 
Sorry, maybe I'm wrong...:nb) but it can be merely defined with an appropriate volume form ( a k-differential form), that is in a k-1 dimens. planes?
 
Jianphys17 said:
Sorry, maybe I'm wrong...:nb) but it can be merely defined with an appropriate volume form ( a k-differential form), that is in a k-1 dimens. planes?

You have to be careful about using the word "plane" in N D. I define a plane as always 2D. A surface is N-1 D. A surface may or may not be planar.

Using this definition, rotation is always in orthogonal planes.
 
Last edited:
  • Like
Likes   Reactions: Jianphys17

Similar threads

Graduate Vector calculus in higher dimensions
  • · Replies 8 ·
Replies
8
Views
7K
Graduate Generalizing: Curl of a Function
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Why Can't the Del Operator Be Treated as a Vector in Curl Calculations?
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad What would a calculus author have to say on ##\int r^2dm##?
  • · Replies 11 ·
Replies
11
Views
3K
Undergrad Derivatives of Standard Functions
  • · Replies 17 ·
Replies
17
Views
2K
Undergrad Linear Algebra and Identity Operator Generalized to 3D
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Vector Derivative: Unifying Calculus on Manifolds and Complex Analysis
  • · Replies 10 ·
Replies
10
Views
6K
High School How can hyperreal numbers make infinitesimals logically sound in calculus?
  • · Replies 24 ·
Replies
24
Views
7K
Trying to review linear algebra after graduating
  • · Replies 16 ·
Replies
16
Views
3K
Vector Operations: Solving Gradient, Divergence, Curl and Laplacian Problems
  • · Replies 4 ·
Replies
4
Views
2K