By Anthony E. Armenàkas

ISBN-10: 1420057774

ISBN-13: 9781420057775

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 method 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 parts of aRead more...

summary: CARTESIAN TENSORS Vectors Dyads Definition and principles 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 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 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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**Example text**

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. 122) This transcendental equation has two solutions in the interval 0 # by 90o ( and # B which differ + B/2).

82) In many problems in mechanics, it is necessary to establish the stationary values of a function f (x1, x2, x3), when the variables are related by a constraining relation g(x1, x2, x3) = 0. If the relation g(x1, x2, x3) = 0 can be solved for one of the variables, say x1, in terms of the other two, then the resulting expression may be substituted into the function f(x1, x2, x3) and a function is obtained. 84) However, in certain problems, g(x1, x2, x3) = 0 is a complicated function. In this case in order to establish the stationary values of f (x1, x2, x3), it is convenient to use the ingenious method proposed by Lagrange, which we describe in the sequel.

89). Thus, the stationary values of f (x1, x2, x3) subjected to the constraining relation g(x1, x2, x3) = 0 and the stationary values of F(x1, x2, x3) without a constraining relation occur at the same points. 92). The multiplying constant 8 is called the Lagrange multiplier. Example 3 Using the method of Lagrange multipliers, find the point on the plane specified by the following relation which is the nearest to the origin of the system of axes x1, x2, x3: (a) Solution The distance d(x1, x2, x3) of any point P (x1, x2, x3) from the origin is given as 26 Cartesian Tensors (b) Thus, we must establish the point at which the function d 2 assumes a stationary value under the constraining relation (a).