I Transformation of second rank tensor

[LightPhoton]

In all the sources I checked, except for one by Dwight E. Neuenschwander, Tensor Calculus for Physics: A Concise Guide, they only provide the definition of a tensor (or describe how it transforms). However, Neuenschwander attempts to motivate the definition of a second-rank tensor.

First, he defines the elements of the transformation matrix as

$$\Lambda_{\alpha i}=\langle \alpha\vert i \rangle= \frac{\partial x'^{\alpha}}{\partial x^i}$$


Then, Neuenschwander takes a general tensor, $\tilde I$, and represents it in the "Greek" basis as a transformation of tensor components in the "Latin" basis:

$$\langle \alpha\vert \tilde I\vert \beta \rangle=\langle \alpha\vert i\rangle\langle i\vert \tilde I\vert j\rangle\langle j\vert \beta \rangle \\ =\Lambda^{\alpha i}I^{ij}\Lambda^{j\beta}$$

At this point, Neuenschwander writes $$\Lambda^{j\beta}={\partial x'^{\beta}}/{\partial x^j}$$, but shouldn't it be $$\Lambda^{j\beta}={\partial x'^{j}}/{\partial x^\beta}?$$

 

[anuttarasammyak]

How he cares high-low position of indexes? The first formula LHS has low but others have high ones.

 

[Ibix]

Indeed. The index placement seems odd to me - in the notation I'm familiar with the transformation matrix should have one low and one high index, and you contract the lower index with the upper index of the vector. He can get away with not caring about upper versus lower in Euclidean geometry, but then I don't understand why he's making a distinction between upper and lower indices at all.

I think the answer to your question depends on how he's using notation. Are you sure you've reproduced it correctly? If so, what rules for index placement has he specified?

 

[LightPhoton]

Indeed. The index placement seems odd to me - in the notation I'm familiar with the transformation matrix should have one low and one high index, and you contract the lower index with the upper index of the vector. He can get away with not caring about upper versus lower in Euclidean geometry, but then I don't understand why he's making a distinction between upper and lower indices at all.

I think the answer to your question depends on how he's using notation. Are you sure you've reproduced it correctly? If so, what rules for index placement has he specified?

He just said to ignore the upper and lower index for the time being

 

[Ibix]

He just said to ignore the upper and lower index for the time being

Unfortunately, index placement is germane to your question, at least in the notation I'm familiar with. It's always possible Neuenschwander is using some convention I'm not familiar with, so read the following with caution.

My guess is that Neuenschwander has tried to simplify by ignoring upper/lower index placement, which mostly works as long as you stick with Cartesian coordinates on flat space. I think he's been slightly bitten by that simplification here.

Using Neuenschwander's Latin-to-Greek convention, to transform an upper index tensor in Latin basis $V^i$, to the Greek basis you contract with $\Lambda^\alpha{}_i=\partial x'^\alpha/\partial x^i$. So$$V^\alpha=\Lambda^\alpha{}_iV^i=\frac{\partial x'^\alpha}{\partial x^i}V^i$$To transform a lower index tensor $V_i$ you contract with $\Lambda_\alpha{}^i=\partial x^i/\partial x'^\alpha$, so$$V_\alpha=\Lambda_\alpha{}^iV_i=\frac{\partial x^i}{\partial x'^\alpha}V_i$$Note that there are different conventions on index order for $\Lambda$, but it's the up/down placement that really matters.

This transformation rule holds for more general tensors - you contract the upper indices with the partial one way up and lower indices with the partial the other way up. For example$$
\begin{eqnarray*}
T^\alpha{}_\beta{}^\gamma&=&
\Lambda^\alpha{}_i
\Lambda_\beta{}^j
\Lambda^\gamma{}_k
T^i{}_j{}^k\\&=&
\frac{\partial x'^\alpha}{\partial x^i}
\frac{\partial x^j}{\partial x'^\beta}
\frac{\partial x'^\gamma}{\partial x^k}
T^i{}_j{}^k
\end{eqnarray*}$$

So if Neuenschwander genuinely means that $I$ is a (2,0) tensor (i.e. both upper indices) then both the lambdas should have primed-and-Greek on the top and unprimed-and-Latin on the bottom. If it's a (0,2) tensor (both lower indices) then both lambdas should have primed/Greek on the bottom and unprimed/Latin on the top. If it's a (1,1) tensor then it should be one of each. The $\langle i|I|j\rangle$ notation makes me suspect it's a (1,1) tensor, but I'm not too familiar with that notation (I mostly use tensors in a GR context where everybody just uses index notation), but the placing of the partials in his definition of $\Lambda^{j\beta}$ suggests a (2,0) tensor. So I'm not sure.

Do note that Neuenschwander's version has correctly associated the Greek index with the prime, which yours hasn't.