By Zlatko Jankocic
Read or Download A Contribution to the Vector and Tensor Analysis: Course Held at the Department for Mechanics of Deformable Bodies September – October 1969 PDF
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Extra info for A Contribution to the Vector and Tensor Analysis: Course Held at the Department for Mechanics of Deformable Bodies September – October 1969
C HA PT ER VI Remarks We add a few remarks to the described approach to the vector and tensor algebra. a) The notation introduced in this paper could be regarded as a contribution to the "direct" system of notation in the sense of Schauten ( 3) . Its usefulness for the vector and 39 Chap. VI - Remarks tensor analysis will be shown in a separate paper. b) The results of Chapter III and the related matter m Chapter V dealing with the connections of the bra and ket forms of vectors and tensors may be regarded as a contribution to the theory of "hybrid quantities" in the sense of Schauten [ 3] • c) We see from Chapter I that the assumptions (1.
In spite of the fact that we developed the described scheme for finite-dimensional vector spaces X(n) it can usefully be applied in a discussion of more general cases, i. e. in infinite-dimensional spaces X (oo) and X (C) He re we do not give a full and rigorous report but indicate Vector and Tensor Algebra 42 some plausible changes tobe made with some illustrations for application. First, we have to interpret the summation over double indices as sums with an infinite number of terms for n =- oo and as integrals over double in- dices for the continuum.
E. ~ (6. lOa) 45 Chap. VI - Remarks ' and the analogous relations for the ket vector forms. W e point out that the scalar product is invariant . > If case b) X> b (6. lOb) is taken into consid- eration, from the relations (6. 6) itfollows ~. (X) • ~ The functions (6. 11) represent an infinite orthorrormal ~~(X) complete set of functions as shown frorn (6. 8a} when written in the conventional rnanner (6. f. o. ' r = d (X- (6. Sb} Y) l is the inte rval of the variable X 46 Vector and Tensor Algebra In case b) the relations (6.