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

In summary, the traditional curl pseudovector can be generalized to more dimensions through the use of a geometric algebra, which is similar to differential forms.
  • #1
Jianphys17
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  • 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:
 
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  • #2
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.
 
  • #3
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)
 
  • #4
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.
 
  • #5
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?
 
  • #6
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.
 
  • #7
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What is the generalized curl operator?

The generalized curl operator, also known as the curl operator or simply curl, is a mathematical operation in vector calculus that describes the rotation of a vector field in three-dimensional space. It is represented by the symbol ∇ × and is commonly used in the study of fluid dynamics, electromagnetism, and other fields of physics and engineering.

How is the generalized curl operator calculated?

The generalized curl operator can be calculated using the determinant of a 3x3 matrix, known as the Jacobian matrix, whose columns are the partial derivatives of the vector field with respect to each coordinate. The resulting vector represents the magnitude and direction of the rotation at each point in the vector field.

What is the physical interpretation of the generalized curl operator?

The generalized curl operator has a physical interpretation as the circulation of a vector field per unit area. This means that it measures the amount of rotation of a vector field around a closed loop, and is therefore useful for studying the movement of fluids and the behavior of electromagnetic fields.

What are some applications of the generalized curl operator?

The generalized curl operator has numerous applications in physics and engineering. It is used in the study of fluid dynamics to understand the behavior of vortices and turbulent flows. In electromagnetism, it is used to calculate the magnetic field around a current-carrying wire or a moving charge. It is also used in image processing to detect edges and in computer graphics to create realistic simulations of fluid and smoke movements.

How does the generalized curl operator relate to other mathematical concepts?

The generalized curl operator is closely related to other mathematical concepts such as the gradient and divergence operators. Together, they form the fundamental theorem of vector calculus, which relates the behavior of a vector field to its underlying source or sink. The generalized curl operator is also related to the Laplace operator, which is used to study the behavior of scalar fields in three-dimensional space.

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