00058___f31b69de994886f5d0f9cfbacda9f570

00058___f31b69de994886f5d0f9cfbacda9f570 - Then tx3 el = J...

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Sec. 2.3 EUCLIDEAN METRIC TENSOR 37 Then x1 = el COS~~, el = J (a)2 + (my, t" 2 2 911 = cos 82 +sin O2 = 1, g12 = (cos02)(-e1sine1) + (sine2)(& COSO~) = o = g21, g22 = (-el sin 02)~ + (el COS~~)~ = (el)2 . P2.3 The line element is ds2 = + (e1)2(de2)2. 2.4. Let 51, x2,53 be rectangular Cartesian coordinates and &,&, 193 be spherical polar coordinates. (See Fig. P2.4.)
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Unformatted text preview: Then tx3 el = J (21)2 + ( 5 2 ) 2 + (23)2 , 5 3 ez = cOs-l J ( X l ) 2 + ( 5 2 ) 2 + (23)2 ' and the inverse transformation is z1 = el sin O2 cos B3 , xz = 81 sin O2 sin O3 , Show that the components of the Euclidean metric tensor in the spherical polar coordinates are and all other gij = 0. The square of the line element is, therefore, ds2 = (dB1)2 + (81)2(d02)2 + (81)2(sin82)2(de3)2....
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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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