Curl of Tensor

1. Sep 25, 2006

touqra

Can we curl a stress tensor? What physically meaning will it be?

2. Sep 25, 2006

chroot

Staff Emeritus
The term "curl" usually applies to vector fields. If there is an equivalent definition of curl for tensor fields, I am not familair with it.

- Warren

3. Sep 25, 2006

robphy

Are you referring to an operation like
$$\nabla_{[a}T_{b]c$$?

4. Sep 25, 2006

touqra

I was thinking of something like Helmholtz's theorem, where if you specify the div and curl of a vector field, you then know everything there is to know about the field.
Maybe there's something similar for rank 2 tensor, like the stress tensor, or higher tensors.

5. Oct 3, 2006

Thrice

Incidentally, the defn of curl resembles the antisymmetrized derivative (F in electromagnetism & the curvature tensor in GR). That's not accidental, is it?

6. Nov 17, 2006

nike^^

Hi

i think that the rotor (curl) of a bilinear tensor T can be defined as follows:

let [T] be the matrix associated with T :

t11 t12 t13
[T] = t21 t22 t23
t31 t32 t33

interpreting the raws of [T] as vectors

T1=t11*e1+t12*e2+t13*e3 => [T1]'=(t11,t12,t13)
T2=t21*e1+t22*e2+t23*e3 => [T2]'=(t21,t22,t23)
T3=t31*e3+t32*e2+t33*e3 => [T3]'=(t31,t32,t33)

we can write [T] as

[T1]'
[T] = ( [T2]' )
[T3]'

then, the rotor (curl) of T is simply :

[rotT] = ( [rotT1] , [rotT2] , [rotT3] )

where [rotT1] , [rotT2] , [rotT3] are the column matrix of the rotor (curl) of vectors T1, T2 and T3

7. May 8, 2007

Norman Albers

Indeed, Thrice, this is not accidental. I am learning much of the nature of axial vectors, curl, and the Minkowski tensor. I need to understand the forms expressed in spherical terms and fields for magnetism.

8. May 17, 2007

Chris Hillman

Curl and all that

There are various things one could mean by this, but yes, there are various ways of generalizing the "curl" from vector calculus. In the context of gtr, one particularly useful formalism involves "curl" and "div" operations on hyperslices using the induced connection in the slice.

Yes indeed, the Hodge decomposition, which applies to p-forms, generalizes the Helmholtz decomposition. This can be stated in various ways: one statement is that any exterior form on a compact boundaryless Riemannian manifold can be uniquely decomposed (as an orthogonal direct sum) as the sum of an exact form, a coexact form, and a harmonic form: $\beta = d\alpha + \delta \gamma + \eta$, where $\alpha$ is a coclosed (p-1)-form, $\gamma$ is a closed (p+1)-form, and $\eta$ is a harmonic p-form (thus, both closed and coclosed).