
#1
Sep2506, 11:12 AM

P: 284

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




#2
Sep2506, 03:41 PM

Emeritus
Sci Advisor
PF Gold
P: 10,424

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
Sep2506, 03:44 PM

Sci Advisor
HW Helper
PF Gold
P: 4,108

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



#4
Sep2506, 06:47 PM

P: 284

Curl of Tensor
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
Oct306, 05:41 PM

P: 232





#6
Nov1706, 12:09 PM

P: 1

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
May807, 08:14 AM

P: 21





#8
May1707, 08:02 PM

Sci Advisor
P: 2,341

Undergraduate courses which mention the Helmholtz decomposition typically omit mention of the harmonic term by adding the additional assumption that the form to be decomposed asymptotically vanishes far from the origin (of R^3), whence by Liouville's theorem we expect the harmonic form appearing in the decomposition to vanish, which it does. See for example Frankel, Geometry of Physics. Exterior forms are antisymmetric tensors, so this won't apply directly to a symmetric tensor. Also, Hodge theory works best in Riemannian manifolds, not Lorentzian manifolds. 


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