By Zlatko Jankocic
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Additional info for A Contribution to the Vector and Tensor Analysis: Course Held at the Department for Mechanics of Deformable Bodies September – October 1969
We illustrate this fact by the exarnple (5. 3 <4
( ) z;:. T - <3 ( \ <4) ( \ ) J (''C() IX -valence. In an analogaus way rnean the lowering of the o<. valence. The relation between the tensor cornponents (5. 8) follows irnrnediately. The connection between the bra and ket vector forrns (Chapter III) can be extended to tensors, and a one-to-one correspondence can be established between the tensors in the product vector space and in its rnirror-irn- Vector and Tensor Algebra 36 age vector space (case A) or mirror-image vector space with changed character (up and down) of the valences of every factor space (case B).
The one-to-one correspondence of the vectors produced by the described linear operators (1. la} with the properties a}, b}, c} we call the parallel displacement Vector and Tensor Analysis I 8 from one point to the other point, the operators themselves are called the parallel displacement operators and the vector Q ( p)G ( a (Q-)p) P from the point a (p) ( a the parallelly displaced vector Q ( P) . 5~- L l The four sets of the indices i. m n2 ' io. 5. cr~ )K o. I - /rJ~< j i I ~- ; (the index k dx dx k k ' .
Based on this feature, is vector and tensor analysis for vector and tensor fields built up in the present paper, the n-dimensional vector spaces being connected with the points of an m-dimensiona l param ete r manifold. )' Q c/' + d XI<) of the manifold. The definitions of the absolute, covariant and parallel displacement differentials (derivatives) follow, the re spectiv~ sum and product rule s having the same structure as those for ordinary differentials (partial derivatives). The absolute, covariant and parallel displacement differentials (derivatives) of every field quantity are easily determined after the absolute differentials of the basis vectors and of the scalar functions of coordinates have been given explicitly.
A Contribution to the Vector and Tensor Analysis: Course Held at the Department for Mechanics of Deformable Bodies September – October 1969 by Zlatko Jankocic