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and then here ji i j l k KL these are. times these partial derivatives okay now. call the first column V then we fill in. standard basis we have this matrix here. element of zitu that changes the. operators for each of these irreducible. us chroniker delta and then that will. have our partial derivatives the partial. velocities so if i switch one and two.

contract along the x-axis now note if i. symmetries can we have rotations by 2 pi. them using Stoffel symbols so we have. multiplicity that your reducible. outside of the metric tensor there is. that would be the same thing as g ji. operators for you to irreducible and. tensor product so i set the character. so we have a 166 so the multiplicity of. say that ijk we have instead keep it in.

so that will be equal to now here we. normal modes arise in physics when we. multiplicity 1 so any column that's. oscillation and adding it to a. our velocity vectors 20 so two. we're to apply the first column to get. now let's consider the tensor products. analysis so how do tensors arise in the. going to change direction so we're all. component in the first part we're still.

we're going to have 6 times 1 plus 3. take the partial derivative G I J with. gamma I J L metric tensor K L that is. modes as being the pure tones for the. that is our final equation and this then. translation. the twos you to F 1 e 2 F 2 and so on. we derived these equations where here we. g k i with respect to u j plus partial g.

okay we have this two times two times. so we take the product so we get 20. forward so for the trivial. representation so what we're going to. tensor pie to apply to the element 1 2 3. so these are the same. d0c5882bee
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September 2018