00079___d04c9c52e56b51e17a1c5ca828cb5940 - shaped Whichever...

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58 TENSOR ANALYSIS Chap. 2 Similarly, using the relations of the contravariant base vectors given in Eq. (21), we obtain (29) AT, = p;.m~~nAm, and A, = pr,/3s,ATs If both coordinate systems are rectangular Cartesian, Eqs. (27), (28) and (29) becomes These formulas tell us how base vectors and tensor components are changed in coordinate transformation. Coordinate transformation occupies a uniquely important position in mechanics for many reasons. One reason is exemplified by Einstein’s using it to develop the theory of relativity. Another reason is that the shape of a natural object of interest to science and engineering often has a natural, preferred coordinate system for its description, e.g., the earth is round, the rail is straight, and the egg is egg-
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Unformatted text preview: shaped. Whichever be the reason, we often find it desirable to transform an equation written in one coordinate system to one valid in another cordinate system. For these tasks the method illustrated above can be helpful. 2.15. GEOMETRIC INTERPRETATION OF COVARIANT DERIVATIVES Consider now a vector field v defined at every point of space in a region R. Let the vector at the point P(0’,02,03) be (1) v(P) = .i(P)g;(P). At a neighboring point P’(0’ + do1, tJ2 + do2, 03 + do3), the vector becomes v(P’) = v(P) + dv(P) = [.“P) + dvi(P)][gz(P) + dgi(P)]. On passing to the limit d0; + 0, we obtain the principal part of the difference and the derivative...
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