Applications of Tensor Analysis by A. J. McConnell

Author:A. J. McConnell
Language: eng
Format: epub
Publisher: Dover Publications, Inc.
Published: 1957-07-14T16:00:00+00:00

If it is satisfied in any one coordinate system it is satisfied in every other. If the axes are Cartesian we have seen that the gmn are constants and therefore the Christoffel symbols in such a system are all zero, that is, (11) reduces in this case to

from which it follows that the components of Xr in Cartesian coordinates are all constants. Consequently the Xr form a parallel field along G and (11) is the equation that a covariant parallel vector-field along C must satisfy.

We next take a contravariant vector-field Xr along C and we leave the student to prove in exactly the same manner, by taking an arbitrary covariant parallel field Ar, that

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