00078___3d6d2c2ad7ae7e9f381714b720d7b077 - Sec. 2.14 57 G...

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Sec. 2.14 GEOMETRIC INTERPRETATION OF TENSOR COMPONENTS 57 together with v = i$g' = vmgm, we obtain (22) A Ur = (g, . gm)vm = Pirnvm , ZI, = (g' ' gm)v, = py,v,. If both coordinate systems are rectangular Cartesian coordinates, gi = and = gi. There is no need to distinguish the superscripts and the subscripts of all quantities and g's, g's are unit base vectors. Then Eq. (14) reduces to (23) Prm = cos(gr, gm) = gr * gm = gm gr 1 that is the direction cosine of the angle between g, and g,. Equa- tion (18) becomes (24) PrkPmk = PkrPkm = 6rm i.e., the transpose of the Cartesian tensor prm is the inverse of Prm. Equa- tions (19), (20) and become (25) = (gr * = Prrnvm om = . gm)G = PrmG * Note that in general we still have /Irm # Pmr for T # m. If @'s and v's are the coordinates of the two rectangular Cartesian co- ordinate systems, Eq.
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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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