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PSEUDO-OBJECTS 91
In a left-handed system, X B points downward, and consequently
(h x B). c < 0. (4.124)
Interpreted in this way, the volume of a parallelogram is a pseudoscalar.
4.6.3 Pseudotensors
Pseudotensors are defined just as you would expect. Upon transformation, the com- ponents of a pseudotensor obey
which is exactly the same as a regular tensor, except for the laid,, term. Again we turn to the cross product to find a good example. Consider two coordinate systems. One, the unprimed system, is a right-handed system, and the other, with primed coordinates, is left-handed. Using the Levi-Civita symbol in both coordinate systems to generate the cross product of A and B gives the relation
The minus sign occurs because, as we showed earlier, the physical direction of the cross product is different in the two coordinate systems. Now the transformation properties of regular vectors can be used to find the relationship between E i j k and &. Because A, B, and the basis vectors are all regular vectors, they transform according to Equation 4.25. Writing the primed components of these vectors in terms of the unprimed, Equation 4.126 becomes
_ -
92 INTRODUCTION TO TENSORS
This expression is true for arbitrary a and B, so we obtain the result
Eijk = - a r i a s j a t &. (4.128)
Keep in mind this applies only when the two systems have opposite handedness. If both systems have the same handedness, the minus sign disappears. Thus for the general case of arbitrary transformation between two orthonormal systems, the Levi-Civita symbol components obey
Eijk = blder%%jatk'&t. (4.129)
Consequently,the Levi-Civita symbol is a pseudotensor.
EXERCISES FOR CHAPTER 4
Determine the transformation matrix [a] that corresponds to a rotation of a two-
dimensional Cartesian system by 30". Obtain the inverse of this matrix [a]-' and
evaluate the following operations:
(a) a;'ajm.
(b) a i j ' a , j. (c) U j ' Q m j. Consider the transformation from a standard two-dimensional Cartesian system ( x l , xz) to a primed system (xi, xi) that results from a reflection about the x1 -axis, as shown below:
1 i I
.I
Express the coordinates of a point in the primed system in terms of the coordinates of the same point in the unprimed system. Determine the elements of the transformation matrix [a] that takes vector components from the unprimed system to components in the primed system. Determine [a]-' in three ways: i. By inverting the [a] matrix found in part (b).
ii. By inverting the coordinate equations of part (a).
iii. By simply switching the primed and unprimed labels on the coordinates.
