00055___7718a27efc043997d7bf57ea8baaa4bd - are admissible...

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34 TENSOR ANALYSIS Chap. 2 and vice versa, it is sufficient to meet the following conditions: (a) The functions fi are single-valued, continuous, and possess (b) The Jacobian determinant J = I&i/dxjl does not vanish at any continuous first partial derivatives in the region R, and point of the region R, i.e., (3) Coordinate transformations with the properties (a), (b) named above are called admissible transformations. If the Jacobian is positive every- where, then a right-handed set of coordinates is transformed into another right-handed set, and the transformation is said to be proper. If the Jacobian is negative everywhere, a right-handed set of coordinates is trans- formed into a left-handed one, and the transformation is said to be improper. In this book, we shall tacitly assume that our transformations
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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.

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