By Thomas T.Y.
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Extra resources for On the existence of integrals of Einsteins gravitational equations for free space and their extension to n variables
Let the new coordinates u1 ; u2 ; u3 be de®ned by specifying the Cartesian coordinates (x1 ; x2 ; x3 ) as functions of (u1 ; u2 ; u3 : x1 f u1 ; u2 ; u3 ; x2 g u1 ; u2 ; u3 ; x3 h u1 ; u2 ; u3 ; 1:54 where f, g, h are assumed to be continuous, dierentiable. A point P (Fig. 16) in space can then be de®ned not only by the rectangular coordinates (x1 ; x2 ; x3 ) but also by curvilinear coordinates (u1 ; u2 ; u3 ). If u2 and u3 are constant as u1 varies, P (or its position vector r) describes a curve which we call the u1 coordinate curve.
Alternatively dV can be written as @ x1 ; x2 ; x3 @r @r @r du du du ; Á Â du du du dV @u1 @u2 @u3 1 2 3 @ u1 ; u2 ; u3 1 2 3 where @x1 @u1 @ x1 ; x2 ; x3 @x2 J @ u1 ; u2 ; u3 @u1 @x3 @u1 @x1 @u2 @x2 @u2 @x3 @u2 1:57 @x1 @u3 @x2 @u3 @x3 @u3 is called the Jacobian of the transformation. 54) is one to one in the neighborhood of a point. We are now ready to express the gradient, divergence, and curl in terms of u1 ; u2 , and u3 .
Tensors of second rank From two contravariant vectors A and B we may form the N 2 quantities A B . This is known as the outer product of tensors. These N 2 quantities form the components of a contravariant tensor of the second rank: any aggregate of N 2 quantities T which, under a coordinate change, transform like the product of 48 BASIC OPERATIONS WITH TE NS ORS two contravariant vectors T @x @x H T ; @x H @x H 1:97 is a contravariant tensor of rank two. We may also form a covariant tensor of rank two from two covariant vectors, which transforms according to the formula T @x H @x H H T : @x @x 1:98 Similarly, we can form a mixed tensor T of order two that transforms as follows: T @x @x H H T : @x H @x 1:99 We may continue this process and multiply more than two vectors together, taking care that their indexes are all dierent.