Tensor Transformations
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TENSOR TRANSFORMATIONS for CURVILINEAR COORDINATES General convention: Unprimed to primed satisf ies e j = a ij e i '. 1. Cartesian to Curvilinear Note that e j = x j = ∂x j h i ∂q i q i , using the curvilinear form of the gradient. Therefore, e j = a ij q i ', where a ij = ∂x j h i ∂q i . 2. Curvilinear to Cartesian Observe q j = q j | q j | = 1 | q j | ∂q j ∂x i e i = 1 | q j | ∂q j ∂x i ∂x i h k ∂q k q k from part 1 above = 1 | q j | ∂q j h k ∂q k q k by the Chain Rule = 1 | q j | 1 h j q j because ∂q j ∂q k = jk Therefore we see that 1 | q j | = h j . So, q j = h j ∂q j ∂x i e i . I.e. q j = b ij e i ', where b ij = h j ∂q j ∂x i . 3. Curvilinear to Curvil inear ' q j = h j ∂q j ∂x k e k = h j ∂q j ∂x k ∂x k h i ' ∂q i ' q i ' = h j ∂q j h i ' ∂q i ' q i ' . Thus, q j = c ij q i ', where c ij = h j ∂q j h i ' ∂q i ' . Finally, because we're using or thonormal bases , we have c ij = h i ' ∂q i ' h j ∂q j .
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Tensor Transformations
Transcript of Tensor Transformations
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TENSOR TRANSFORMATIONS
for
CURVILINEAR COORDINATES
General convention: Unprimed to primed satisfies ej = aij ei'.
1. Cartesian to Curvilinear
Note that ej = xj = xj
hiqi qi , using the curvilinear form of the gradient.
Therefore, ej = aij qi', where aij = xj
hiqi .
2. Curvilinear to Cartesian
Observe qj = qj|qj|
= 1
|qj| qjxi
ei = 1
|qj| qjxi
xi
hkqk qk from part 1 above
= 1
|qj|
qjhkqk
qk by the Chain Rule
= 1
|qj| 1
hj qj because
qjqk
= jk
Therefore we see that 1
|qj| = hj . So, qj =
hj qjxi
ei .
I.e. qj = bij ei', where bij = hj qjxi
.
3. Curvilinear to Curvilinear '
qj = hj qjxk
ek = hj qjxk
xk
hi ' qi ' qi' =
hj qjhi ' qi '
qi' .
Thus, qj = cij qi', where cij = hj qj
hi ' qi ' .
Finally, because we're using orthonormal bases, we have cij = hi ' qi '
hj qj .
