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Chapter 10: Q7P (page 517)
Write the transformation equations forto verify the results of Example 3.
This answer proves that is a polar vector.
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In equationlet the variables be rectangular coordinates x, y, z, and let , be general curvilinear coordinates, orthogonal or not (see end of Section 8 ). Show that is the matrix in [or in for an orthogonal system]. Thus show that the volume element in a general coordinate system is where , and that for an orthogonal system, this becomes [by or ], . Hint: To evaluate the products of partial derivatives in , observe that the same expressions arise as in finding . In fact, from and , you can show that row i times column j in is just in equations to .
Write the tensor transformation equations for to show that this is a (rank 6) tensor (nota pseudo tensor). Hint:Write (6.1) for eachand multiply them, being careful not to re-use a pair of summation indices.
Let be the tensor in . This is a -rank tensor and so has components. Most of the components are zero. Find the nonzero components and their values. Hint: See discussion after .
Write the transformation equations to show that is a pseudo vector if Vis a vector. Hint:See equations (5.13), (6.2), and (6.3).
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