lec10-3 - and E j = a kj E k Transformations continued •...

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Transformation of tensors Rather than simply a matrix of numbers, tensors depend on the definition of a coordinate system The physics does not depend on the coordinate system, so we need well-defined rules to describe how tensors transform under coordinate transformations We will deal here explicitly only with Cartesian tensors (we could have tensors defined in other ways, for example in a spherical coordinate system, etc)
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Transformation of Cartesian tensors Consider two rank 1 tensors related by a rank 2 tensor, for example the conductivity tensor J i = σ ij E j Now let’s consider a rotated coordinate system, such that we have the transformations x 0 i = a ij x j We showed in class how the a ij are related to direction cosines between the original and rotated coordinate system Then we should have J 0 i = a ij J j , and E 0 j = a kj E j Recall that for orthogonal transformation, A T = A - 1 , so J i = a ji J 0 j
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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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lec10-3 - and E j = a kj E k Transformations continued •...

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