Thanks for the help on the other questions.(adsbygoogle = window.adsbygoogle || []).push({});

I am having trouble with another derivation. Unlike the others, it's not abstract whatsoever.

Okay I wish to find the transformation Law for the components of a rank 2 tensor.

Easy, I know: [tex]T: V^* \times V \mapsto \mathbb{R}[/tex]

So

[itex]T = T^i_{\phantom{i} j} e_i \otimes e^j[/itex]

I wish to find

[itex]T^i'_{\phantom{i'} j'}[/itex]

Where the following hold:

[tex]e_{i'}=a^{J}_{\phantom{J} i'} e_J[/tex]

and

[tex]e^{i'}=b^{i'}_{\phantom{i'} J} e^J[/tex]

where, the coefficients are just real numbers. Now [itex]T^i_{\phantom{i} j}=T(e^i, e_j)[/itex] so that:

[itex]T^{i'}_{\phantom{i'} j'}=T(e^{i'}, e_{j'})[/itex]

[itex]=T(b^{i'}_{\phantom{i'} J} e^J, a^{L}_{\phantom{L} j'} e_L)[/itex]

By linearity of [itex]\otimes[/itex] we have:

[itex]=b^{i'}_{\phantom{i'} J} a^{L}_{\phantom{L} j'} T( e^J, e_L)[/itex]

[itex]=b^{i'}_{\phantom{i'} J} a^{L}_{\phantom{L} j'} T^{J}_{\phantom{J} L}[/itex]

Now my lecturer done a funny thing and said,

"it may be shown that [itex]b=a[/itex]"

which confuses the hell outta me because,

[tex]e_{i'}=a^{J}_{\phantom{J} i'} e_J[/tex]

[tex]e^{i'}=b^{i'}_{\phantom{i'} J} e^J[/tex]

abide by the duality relation so that:

[tex]e^{i'} e_{j'}=b^{i'}_{\phantom{i'} J} e^J a^{K}_{\phantom{K} j'} e_K=\delta^{i'}_{j'}[/tex]

So do the original bases obey their own set of duality relations so that:

[tex]e^{i'} e_{j'}=b^{i'}_{\phantom{i'} J} a^{K}_{\phantom{K} j'} \delta^J_K=\delta^{i'}_{j'}[/tex]

So

[tex]e^{i'} e_{j'}=b^{i'}_{\phantom{i'} K} a^{K}_{\phantom{K} j'} =\delta^{i'}_{j'}[/tex]

Is this not the definition of [itex]a[/itex] being the inverse of [itex]b[/itex]. Of course, I know a priori I am wrong as this would give [itex]T^{i'}_{\phantom{i'} j'}=T^{i}_{\phantom{i} j}[/itex] regardless of transformation.

In my definition of [itex]a[/itex] and [itex]b[/itex], the superindex refers to row and the lower index refers to column.

My gut feeling is that I am using the matrix notation all wrong.

Any takers?

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# Transformation of Rank 2 mixed tensor

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