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Unformatted text preview: and E j = a kj E k Transformations continued We can use this transformation, J i = ij E j In terms of the entire transformation matrix A , A T J = A T E Multiply on the left by A , J = A A T E Then we see = A A T , in components ij = a ik kl a jl Covariant and contravariant tensors We might have a vector like ~ V = ~ In a rotated system, ~ V = ~ where the gradient is with respect to the primed system x i = X j x j x i x j This is an example of a covariant vector Contrast to the rotations x i = a ij x j , or for any general vector components V i = a ij V j a ij = x i x j Then from V i = a ij V j we have a contravariant vector V i = X j x i x j V j...
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 Spring '03
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