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Chapter 10: Q9 P (page 517)
F and E are polar vectors.
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Elliptical 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.
The square matrix in equation is called the Jacobian matrix J; the determinant of this matrix is the Jacobian which we used in Chapter 5 , Section 4 to find volume elements in multiple integrals. (Note that as in Chapter 3, J represents a matrix; J in italics is its determinant.) For the transformation to spherical coordinates in localid="1659266126385" and show that . Recall that the spherical coordinate volume element is . Hint: Find and note that
Mass of uniform density=1, bounded by the coordinate planes and the plane x +y +x=1 .
Using (10.15) show thatis a-rank covariant tensor. Hint:Write the transformationequation for each, and set the scalarto find the transformationequation for.
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