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Unformatted text preview: are admissible and proper. 2.3. EUCLIDEAN METRIC TENSOR The first thing we must know about any coordinate system is how to measure length in that reference system. This information is given by the metric tensor. Consider a three-dimensional Euclidean space, with the range of all indices 1, 2, 3. Let (1) ei = q X l , x2, x3) , be an admissible transformation of coordinates from the rectangular Cartesian (in honor of Cartesius, i.e., Descartes) coordinates z1,x2,23 to some general coordinates 81,82,83. The inverse transformation is assumed to exist, and the point ( x ~ , x ~ , Q ) and (&,&, 6 3 ) are in one-to- one correspondence. Consider a line element with three components given by the differ- entials dx', dx2, dx3. Since the coordinates 21, 22, x3 are assumed to be...
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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.
- Fall '05