In Dimensional analysis why is Lenght/Lenght=1 (a dimensionless number)?by ChARMELeOn Tags: analysis, dimensional, dimensionless, lenght or lenght1, number 

#55
Dec2510, 06:25 AM

P: 46

EDIT: He also admits that his method doesn't work for equations with fractional exponents. 



#56
Dec2510, 12:02 PM

P: 3,015

[tex] \hat{\sigma}_{i} \, \hat{\sigma}_{k} = \delta_{i k} \, \hat{1} + i \, \epsilon_{i k l} \, \hat{\sigma}_{l} [/tex] obey a similar algebra as [itex]V[/itex] except that it is antiAbellian for the Pauli matrices that are not equal. Perhaps I need more group theoretical knowledge to refine this point. One might ask, what do Pauli matrices have to do with rotations. Again, there is a very "convenient" coincidence in 3d. BTW, at one point in his first paper (third paragraph on the sixth page), he says that [itex]\sin{(\theta)}[/itex] is orientationally quite different from [itex]\sin{(\phi)}[/itex], where [itex]\theta[/itex] is an angle (with orientational symbol 1_{z}) and [itex]\phi[/itex] is a phase angle. 



#57
Dec2510, 02:33 PM

P: 789

[tex]\sin(\theta\,\,1x+\phi \,\,1x)=\sin(\theta \,\,1x)\cos(\phi\, \,1x)+\sin(\phi \,\,1x)\cos(\theta \,\, 1x) = 1x\, \sin(\theta)\cos(\phi)+1x \,\sin(\phi)\cos(\theta) [/tex] so that [tex]\sin(\theta\,\,1x+\pi/2\,\,1x)=1x \cos(\theta)[/tex] and the discrepancy disappears. [tex](A_j\mathbf{e}_j 1_j)\cdot(B_k \mathbf{e}_k 1_k)=A_j B_j 1_j^2 = A_j B_j 1_0[/tex] which proves [tex]1_j^2=1_0[/tex] (dimensionless). Then do the cross product: [tex](A_j\mathbf{e}_j 1_j) \mathrm{X} (B_k \mathbf{e}_k 1_k)=\varepsilon_{ijk}A_j B_k \mathbf{e}_i (1_j 1_k) = \varepsilon_{ijk}A_j B_k \mathbf{e}_i 1_i[/tex] which proves that [tex]1_j 1_k= 1_i[/tex] where i,j, and k are all different. The same procedure could be carried out for the invariant analogs in relativity for the direction symbols 1x, 1y, 1z, 1t, 10. 



#58
Dec2510, 02:54 PM

P: 3,015

Siano actually talks about this. Due to orientational analysis, we can distinguish between angular velocity (which has orientation) and circular frequency (which does not); torque (which is oriented) and work (which is not) and so on. Perhaps a very drastic example is the case of coefficient of surface tension. It is defined as the energy per unit area and, thus it has the same orientation as the area. It's dimension is, however [itex]\mathrm{M} \mathrm{T}^{2}[/itex], just as the rate of change of growth of, e.g. an animal, which, of course, is orientationless. 



#59
Dec2510, 10:18 PM

P: 789

[tex](\mathbf{A} \mathrm{x} \mathbf{B})_i = \varepsilon_{ijk}A_jB_k[/tex] where [tex]\varepsilon_{ijk}[/tex] is the permutation symbol (=1 for even permutations of 123, 1 for odd, and zero otherwise). For a 4d Euclidean space the analog is [tex] \varepsilon_{ijkl}A_kB_l[/tex] where [tex]\varepsilon_{ijkl}[/tex] is the permutation symbol (=1 for even permutations of 1234, 1 for odd, and zero otherwise). For Minkowski space, we need to make the covariant/contravariant distinction and the analog is [tex] \varepsilon_{ijkl}A^kB^l[/tex] Note that the analog is a 2nd rank tensor rather than a vector. 


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