Someone claims that the array B_ ij = pmatrix 2&3&4 3&4&5 4&5&6 pmatrix (that is

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Action and Symmetry, Ch Non-inertial frames from one Lagrangian

Problem

Someone claims that the array

B_{ij}=\begin{pmatrix}2&3&4\\3&4&5\\4&5&6\end{pmatrix}

(that is,

B_{ij}=i+j

with

i,j=1,2,3

) represents a genuine physical quantity with exactly these same nine components in any set of Cartesian axes. Test the claim for a rotation of axes by

90^\circ

about the

z

-axis, using the Cartesian-tensor transformation rule

T'_{ij}=R_{ik}R_{jl}T_{kl}

with

R=\begin{pmatrix}0&1&0\\-1&0&0\\0&0&1\end{pmatrix},

and state whether the claim is consistent with

B

being a Cartesian tensor.

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