Math 402 Set VI 1. Convert [ ± ² ψ (x,y,z) - λψ (x,y,z) = 0 ] to a "covariant" form which transforms like a tensor under a generalized coordinate transformation, dx i = A i j dq j . λ is a scalar ( a numerical constant and independent of the coordinate system). What is the rank of the tensor and how could you show it transforms accordingly? 2. Find the number of independent components of the fourth rank tensor, M ijm ± , which is antisymmetric with respect of the interchange of any two indices. The indices run from 1 to 3. 3. In the q i coordinate system related to the x j system by dx j = A j k dq k : T' ± m = R' ± m- D' ± m , where T' ± m , R' ± m and D' ± m are tensors under A . Show that in the x j system: T ± m = R ± m- D ± m . 4. Use summation notation to evaluate the following expressions. Put the answer in the simplest form. The ± operator acts on all functions (of x,y,z) to the right of it. Bold face type indicates vector; a is a constant vector. a) [
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