00054___9c09a177778cb4b8e1d9febe3851bcf8

00054___9c09a177778cb4b8e1d9febe3851bcf8 - the components...

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Sec. 2.2 COORDINATE TRANSFORMATION 33 which can be written as The vector product of two vectors x and y is the vector z = x x y whose three components are (14) 21 = x2Y3 - X3Y2 1 z2 = x3Y1 - xlY3 > z3 = XlY2 - z2Yl . This can be shortened by writing PROBLEMS 2.1. Show that, when a, j, k range over 1,2,3, (a) 6ij6ij = 3 (c) eijkAjAk = 0 (b) eijkejki = 6 (d) 6ij6jk = 6ik 2.2. Verify the following identity connecting three arbitrary vectors by means of the e-6 identity. A x (B x C) = (A. C)B - (A. B)C . Note: The last equation is well known in vector analysis. After identifying the quantities involved as Cartesian tensors, this verification may be construed as a proof of the e-6 identity. 2.2. COORDINATE TRANSFORMATION The central point of view of tensor analysis is to study the change of
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Unformatted text preview: the components of a quantity such as a vector with respect to coordinate transformations. A set of independent variables x 1 , 2 2 , 2 3 may be thought of as specifying the coordinates of a point in a frame of reference. A transformation from q r 2 2 , x 3 to a set of new variables 2 1 , 3 2 , 2 3 through the equations specifies a transformation of coordinates. The inverse transformation (2) xi = gi(Z1, 2 2 1 33) i = 1,2,3, proceeds in the reverse direction. In order to insure that such a transforma- tion is reversible and in one-to-one correspondence in a certain region R of the variables ( q , 2 2 , z 3 ) , i.e., in order that each set of numbers (XI, 2 2 , ~ ) defines a unique set of numbers ( 2 1 , 3 2 , 2 3 ) , for (XI, 2 2 , ~ ~ ) in the region R,...
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