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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I Index notation, covector transfor ( matrix representation)

Have something to add?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**