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Unformatted text preview: and E j = a kj E k Transformations continued • We can use this transformation, J i = σ ij E j • In terms of the entire transformation matrix A , A T J = σ A T E • Multiply on the left by A , J = A σ A T E • Then we see σ = A σ A T , in components σ ij = a ik σ kl a jl Covariant and contravariant tensors • We might have a vector like ~ V = ~ ∇ φ • In a rotated system, ~ V = ~ ∇ φ where the gradient is with respect to the primed system ∂φ ∂ x i = X j ∂ x j ∂ x i ∂φ ∂ 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
 Staff
 Tensor, Coordinate system, ej, Covariance and contravariance of vectors, contravariant tensors

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