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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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Verify Hints: In Figure , consider the projection of the slanted face of area onto the three unprimed coordinate planes. In each case, show that the projection angle is equal to an angle between the axis and one of the unprimed axes. Find the cosine of the angle from the matrix A in .
Verify equations(2.6).
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
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" .
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