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Chapter 10: Q11P (page 505)
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Answer
The equation has been proven.
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Show that, in polar coordinates, thecontravariant component of dsis which is unitless, the physical component of ds is which has units of length, and thecovariant component of ds iswhich has units role="math" localid="1659265070715" .
Parabolic cylinder coordinates
Show that the nine quantities (which are the Cartesian components of where V is a vector) satisfy the transformation equations for a Cartesian -rank tensor. Show that they do not satisfy the general tensor transformation equations as in . Hint: Differentiate orpartially with respect to, say,. You should get the expected terms [as in ] plus some extra terms; these extraneous terms show that is not a tensor under general transformations. Comment: It is possible to express the components of correctly in general coordinate systems by taking into account the variation of the basis vectors in length and direction.
Generalize Problem 3 to see that the direct product of any two isotropic tensors (or a direct product contracted) is an isotropic tensor. For example show thatis an isotropic tensor (what is its rank?) andis an isotropic tensor (what is its rank?).
Show that the sum of two -rank tensors is a -rank tensor. Hint: Write the transformation law for each tensor and then add your two equations. Divide out the factors to leave the result .
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