# A Course of Differential Geometry by Campbell J.E. By Campbell J.E.

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Additional resources for A Course of Differential Geometry

Sample text

We also be combined in other ways, as form the tensor derivative of the product of two tensors by the same rule as in ordinary differentiation. The tensors a^ and a ift are called fundamental tensors. We have seen that they have the property of being annihilated by any operator As regards tensor derivation they jp. lore play the part of constants. The symbol e^ satisfies the definition of a tensor. is called a Any tensor, formed by taking the product of a tensor and tensor from which is it is said to be an associate tensor of the derived.

26 . 4) variables x^ We . can therefore take the ground form to be cZs R-*. {221} = (212) a function of 0, + (221) = and therefore We = = we have is zero, (122) so that 35 an integral of the complete system, we have {111} From K IS CONSTANT WHEN REDUCTION OF A GROUND FORM ~a 22 c ^p ot( J) = cot 2a 22 = ; i//-, 5 ) THE GROUND FORM WHEN U = 36 so that sin 2 ( fl) C/ is a function of We may rr only. therefore take the = 2 K or if we sin 2 dxi + x\, -,~ dx'] , = ^/ 2 5 the ground form as = ds 2 When ground form as ^ = Rx\, or is ground form 22 2 (/6- if we take we may take 2 2 (dU-J + sin 2 ^rficS).

Are it tensor components might make the general theorem, : whose proof is rather complicated, more easily understood. The square of the tensor whose components are Ul ... Un If we form the is a tensor whose components are U^U^ associate tensor a^U^U^ we have an invariant which is . 3) . differential parameter. Similarly by forming the tensor which is the product of BELTRAMIS THREE DIFFERENTIAL PARAMETERS 17 two tensors whose components are t^ ... Un and Vl ... 4) . We. also have Beltrami's second differential parameter & 2 (U) = these all Clearly are invariants.