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Kashmir

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chapter 4.8 of Goldstein’s classical mechanics 3rd edition that deals with infinitesimal rotations, and the following is the part I got stuck:

(p.166~167) :

I'm not able to understand what the author is trying to say. How does "If ##d\boldsymbol{\Omega}## is to be a vector in the same sense as ##\mathbf{r}##, it must transform under ##\mathbf{B}## in the same way" prove that ##d\boldsymbol{\Omega}## is indeed a vector?

I appreciate your help.

(p.166~167) :

"If ##d\boldsymbol{\Omega}## is to be a vector in the same sense as ##\mathbf{r}##, it must transform under ##\mathbf{B}## in the same way. As we shall see, ##d\boldsymbol{\Omega}## passes most of this test for a vector, although in one respect it fails to make the grade. One way of examining the transformation properties of ##d\boldsymbol{\Omega}## is to find how the matrix ##\boldsymbol{\epsilon}## transforms under a coordinate transformation. The transformed matrix ##\boldsymbol{\epsilon}’## is obtained by a similarity transformation:

##\begin{equation} \boldsymbol{\epsilon}’=B\boldsymbol{\epsilon}B^{-1}\end{equation}##

As the antisymmetry property of a matrix is preserved under an orthogonal similarity transformation, ##\boldsymbol{\epsilon}’## consists of nonvanishing elements ##d\Omega’_i## such that ##\begin{equation}d\Omega'_i=|B|b_{ij}d\Omega_j.\end{equation}"##"

I'm not able to understand what the author is trying to say. How does "If ##d\boldsymbol{\Omega}## is to be a vector in the same sense as ##\mathbf{r}##, it must transform under ##\mathbf{B}## in the same way" prove that ##d\boldsymbol{\Omega}## is indeed a vector?

I appreciate your help.

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