Einstein Summation Convention

2008 ◽  
pp. 1038-1038
Keyword(s):  
Summation Convention ◽  
Einstein Summation Convention

scholarly journals Pure torsion problem in tensor notation

2019 ◽  
Vol 18 (1) ◽  
pp. 057-069
Author(s):  
Sławomir Karaś
Keyword(s):  
Elasticity Theory ◽  
Reference Frames ◽  
Short Description ◽  
Torsion Problem ◽  
Pure Torsion ◽  
Tensor Calculus ◽  
Classic Problem ◽  
Summation Convention ◽  
Einstein Summation Convention

The paper examines the application of the tensor calculus to the classic problem of the pure torsion of prismatic rods. The introduction contains a short description of the reference frames, base vectors, contravariant and covariant vector coordinates when applying the Einstein summation convention. Torsion formulas were derived according to Coulomb’s and Saint-Venant’s theories, while, as a link between the theories, so-called Navier’s error was discussed. Groups of the elasticity theory equations were used.

scholarly journals The New Approach to Analysis of Thin Isotropic Symmetrical Plates

2020 ◽  
Vol 10 (17) ◽  
pp. 5931
Author(s):  
Mykhaylo Delyavskyy ◽  
Krystian Rosiński
Keyword(s):  
Present Method ◽  
Automatic Differentiation ◽  
Linear Equations ◽  
High Accuracy ◽  
Rectangular Plates ◽  
System Of Linear Equations ◽  
New Approach ◽  
Summation Convention ◽  
Analytical Formulae ◽  
Einstein Summation Convention

A new approach to solve plate constructions using combined analytical and numerical methods has been developed in this paper. It is based on an exact solution of an equilibrium equation. The proposed mathematical model is implemented as a computer program in which known analytical formulae are rewritten as wrapper functions of two arguments. Partial derivatives are calculated using automatic differentiation. A solution of a system of linear equations is substituted to these functions and evaluated using the Einstein summation convention. The calculated results are presented and compared to other analytical and numerical ones. The boundary conditions are satisfied with high accuracy. The effectiveness of the present method is illustrated by examples of rectangular plates. The model can be extended with the ability to solve plates of any shape.

Notation

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Pairwise Disjoint ◽  
Higher Order ◽  
Multi Index ◽  
Basic Construction ◽  
Summation Convention ◽  
Index Notation ◽  
Einstein Summation Convention ◽  
Derivatives Of

This chapter explains the notation for the basic construction of the correction. It employs the Einstein summation convention, according to which there is an implied summation when a pair of indices is repeated, and the conventions of abstract index notation, so that upper indices and lower indices distinguish contravariant and covariant tensors. It also presents the notation concerning multi-indices which will later prove helpful for expressing higher order derivatives of a composition. In this notation, a K-tuple of multi-indices is said to form an ordered K-partition of a multi-index if there is a partition whereby the subsets are pairwise disjoint and are ordered by their largest elements.

scholarly journals On Kilmister's Conditions for the Existence of Linear Integrals of Dynamical Systems

1965 ◽  
Vol 14 (3) ◽  
pp. 243-244 ◽  
Author(s):  
R. H. Boyer
Keyword(s):  
Dynamical Systems ◽  
Singular Integral ◽  
First Integral ◽  
Killing Field ◽  
Summation Convention ◽  
Covariant Derivation ◽  
Image Position ◽  
Generally Covariant ◽  
Linear Integrals

Kilmister (1) has considered dynamical systems specified by coordinates q( = 1, 2, , n) and a Lagrangian(with summation convention). He sought to determine generally covariant conditions for the existence of a first integral, , linear in the velocities. He showed that it is not, as is usually stated, necessary that there must exist an ignorable coordinate (equivalently, that b must be a Killing field:where covariant derivation is with respect to a). On the contrary, a singular integral, in the sense that for all time if satisfied initially, need not be accompanied by an ignorable coordinate.

A new notation for quantum mechanics

1939 ◽  
Vol 35 (3) ◽  
pp. 416-418 ◽  
Author(s):  
P. A. M. Dirac
Keyword(s):  
Quantum Mechanics ◽  
Careful Consideration ◽  
Tensor Analysis ◽  
Summation Convention

In mathematical theories the question of notation, while not of primary importance, is yet worthy of careful consideration, since a good notation can be of great value in helping the development of a theory, by making it easy to write down those quantities or combinations of quantities that are important, and difficult or impossible to write down those that are unimportant. The summation convention in tensor analysis is an example, illustrating how specially appropriate a notation can be.

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