00057___a23cffd42a3a5a5a38adac1abcbb0051 - The law of...

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36 TENSOR ANALYSIS Chap. 2 Let &, $2, & be another general coordinate system. Let (9) ei = &(GI, e,, e3) , be the transformation of coordinates from el,&, to el,&, 03. Now Hence, from Eq. (6), Then Eq. (11) assumes a form which is the same as Eq. (3) or (6): (13) ds2 = [email protected]". Accordingly, we call gln the components of the Euclidean metric tensor in the coordinate system el , &,&. The quadratic differential forms (3), (6), and (13) are of fundamental importance since they define the length of any line element in general co- ordinate systems. We conclude that if &,&, 83 and &,&, e3 are two sets of general coordinates, then Euclidean metric tensors gkrn(61, &,e3) and gkm(81,82, 83) are related by means of the law of transformation (12).
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Unformatted text preview: The law of transformation of the components of a quantity with respect to coordinate transformation is an important property of that quantity. In the following section, we shall see that a quantity shall be called a tensor if and only if it follows certain specific laws of transformation. All the results above apply as well to the plane (a two-dimensional Euclidean space), as can be easily verified by changing the range of all indices to 1, 2. P R O B L E M S 2.3. Find the components of the Euclidean metric tensor in plane polar coordinates (0, = r,& = 8; see Fig. P2.3) and the corresponding expression for the length of a line element Ans. Let z1,22 be a set of rectangular Cartesian coordinates....
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This note was uploaded on 01/04/2012 for the course ENG 501 taught by Professor Thomson during the Fall '05 term at MIT.

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