On the Transposition Operators in a Generalised Vector and Tensor Calculus Scheme
In previous papers [1], [2] and [3] we described the basic features of a generalized scheme of the vector and tensor calculus. In the present paper we investigate in detail the properties and role of the transposition operators in this scheme. In Chapter
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I
ZLATKO JANKOVIC UNIVERSITY OF ZAGREB
SELECTED TOPICS AND APPLICATIONS OF TENSOR ANALYSIS
COURSE HELD AT THE DEPARTMENTS FOR MECHANICS OF DEFORMABLE BODIES AND FOB HYDRO-AND GASDYNAMICS JUNE - JULY 1970
UDINE 1970
CO U R S E SAN D
L E C T U RES
-
No.
~2
ISBN 978-3-211-81165-8 ISBN 978-3-7091-2892-3 (eBook) DOI 10.1007/978-3-7091-2892-3
Copyright 1970 by Springer-Verlag Wien Originally published by Springer-Verlag Vienna in 1970 These· lecture notes are manuscript of a paper which will be pub1ished in TENSOR, vo1. 22 (1971) N° 2.
PRE
F ACE
This series of seven lectures entitled "Selected topics in the tensor calculus" is a natural continuation of the lectures
I~
contribution to the
vector and tensor analysis", held at the International Centre for Mechanical Sciences in 1969. The essential points of a general scheme of the vector and tensor calculus described there in detail are briefly reviewed here for further deveZopment of the scheme. The central role in this development is played by the transposition operators which establish the one-to-one correspondence between the bra (ket) and ket (bra) vector spaces. The properties and significance of these operators in the scheme are discussed in full detail. One of the most important conclusions drawn is that the whole calculus can be formed with the help of the notion of absolute differential and is based on the identical vanishing of the absolute differentials of the fundamental and transposition operators. An illustrative example is given to show the generality and efficiency of the proposed scheme. In the author's opinion, the aim of the lecture notes is best realized by the manuscript of his paper "On the transposition operators in a generaZ veotor and tensor calculus scheme", to be published in Tensor N.S.
22 (1971), N.2 which contains a systematic
4 presentation of the above mentioned subject. The author expresses his deepest appreciation and gratitude to the Authorities of particularly to the Secretary General Prof. L. Sobrero and to the Rectors Prof. Prof.
w.
o.
CISM~
Onicescu and
OZszak for giving him this excellent oppor-
tunity to present his recent results of the study of the vector and tensor calculus to such a prominent audience in such a high-level international scientific institution.
z. Zagreb~
July 19?0.
Jankovic
/
5
On the Transposition Operators in a Generalised Vector and Tensor Calculus Scheme
In previous papers [lJ, [2] and
[3]
we
described the basic features of a generalized scheme of the vector and tensor calculus. In the present paper we investigate in detail the properties and role of the transposition operators in this scheme. In Cha£ ter 1 the fundamentals of the scheme are described, while in Chapter 2 the basic properties of the transposition operators are established. In Chapter 3 the consequences of the required identical vanishing of the absolute differentials of the fundamental operator and of the transposition operators are investigated and the coefficients of connection are expressed in terms of
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