Just a couple of quick questions on index notation, may be because of the way I'm thinking as matrix representation:(adsbygoogle = window.adsbygoogle || []).push({});

1) ##V^{u}B_{kl}=B_{kl}V^{u}## , i.e. you are free to switch the order of objects, I had no idea you could do this, and don't really understand for two reasons:

i) In the above say I am in 4-d space and ##V^{u}## is a 4-vector, which can be repesented as a column vector , and ##B_{kl}## can be written as a 4x4 matrix, then the LHS doesn't make sense, you can't multiply, but the right side does.

ii) Or say I have ##A_{mn}B_{kl}## and both of these can be represented as a 4x4 matrix, and matrix multiplication is in general not commutative....

2) I am looking at how a covector transforms and I have:

## w_{u}=\Lambda^{v}_{u} w'_{u}##, where ##w'_{u}## is the transformed covector in some other coordinates ##x'^{u}## rather than ##x^{u}## and ##\Lambda## is the Jacobian transformation of the coordinates ##= \frac {\partial x'}{ \partial x } ## .

Now I want to invert this. I wanted to multiply both sides by ##(\Lambda^{v}_{u})^{-1}##, but I get ## w'_{u}= (\Lambda^{u}_{v}) ^{-1} w_{v}## which I can see straight away is wrong by the inconsistent placement of the indices.

(I am able to derive the correct expression using the chain rule , quite simply, but I'd like to know what is wrong with what I am doing).

Many thanks in advance.

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# I Index notation, covector transfor ( matrix representation)

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