By Anthony E. Armenàkas

CARTESIAN TENSORS Vectors Dyads Definition and principles of Operation of Tensors of the second one Rank Transformation of the Cartesian elements of a Tensor of the second one Rank upon Rotation of the process of Axes to Which they're Referred Definition of a Tensor of the second one Rank at the foundation of the legislation of Transformation of Its elements Symmetric Tensors of the second one Rank Invariants of the Cartesian parts of a Read more...

summary: CARTESIAN TENSORS Vectors Dyads Definition and ideas of Operation of Tensors of the second one Rank Transformation of the Cartesian parts of a Tensor of the second one Rank upon Rotation of the approach of Axes to Which they're Referred Definition of a Tensor of the second one Rank at the foundation of the legislations of Transformation of Its parts Symmetric Tensors of the second one Rank Invariants of the Cartesian elements of a Symmetric Tensor of the second one Rank desk bound Values of a functionality topic to a Constraining Relation desk bound Values of the Diagonal elements of a Symmetric Tensor of the second one

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We denote the components of this tensor with respect to the x1,x2,x3 axes by Aij(i, j = 1, 2, 3)(A13 = A23 = 0). 116) for the quasi plane 34 Cartesian Tensors form of a symmetric tensor of the second rank show that its diagonal and non-diagonal components are continuous functions of the angle angle . For certain values of the , the diagonal components of the tensor assume their stationary values. 116c) for are called its principal values. In order to establish the principal axes of the tensor we set the derivative of AN11 with respect to equal to zero.

116) for the quasi plane 34 Cartesian Tensors form of a symmetric tensor of the second rank show that its diagonal and non-diagonal components are continuous functions of the angle angle . For certain values of the , the diagonal components of the tensor assume their stationary values. 116c) for are called its principal values. In order to establish the principal axes of the tensor we set the derivative of AN11 with respect to equal to zero. 119) is satisfied for any value of . That is, any pair of two mutually perpendicular axes in the x1 x2 plane constitutes, with the x3 axis, a set of principal axes.

72) is often used as the basis for the definition of a tensor of the second rank as an entity which possesses the following properties: 1. With respect to any set of rectangular axes it is specified by an array of nine numbers Aij (i,j = 1, 2, 3) — its nine cartesian components. 22 Cartesian Tensors 2. 72). 3 is not. 75) For instance, the tensor /A whose components with respect to a rectangular system of axes is given as is a symmetric tensor of the second rank. 9 we show that for any symmetric tensor of the second rank, there exists at least one system of rectangular axes x1, x2, x3, called principal, with respect to which the diagonal components of the tensor assume their stationary values.

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