Chapter 1(2)

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INTRODUCTION INTRODUCTION INTRODUCTION INTRODUCTION

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Transcript of Chapter 1(2)

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Example 2.6.2

Example 2.6.1Let T be a nonzero transformation that transforms every vector into a fixed nonzero vector n. Is thistransformation a tensor?

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Example 2.6.2Let T be a transformation that transforms every vector into a vector that is k times the originalvector. Is this transformation a tensor?

Example 2.6.3Consider a transformation T that transforms every vector into its mirror image with respect to afixed plane. Is T a tensor?

Example 2.6.5

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Example 2.6.5Let T be a tensor that transforms the specific vectors a and b as follows:

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Example 2.7.1Obtain the matrix for the tensorT that transforms the base vectors as follows:

Example 2.7.2

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Example 2.7.2Let T transform every vector into its mirror image with respect to a fixed plane; if e1 is normal tothe reflection plane (e2 and e3 are parallel to this plane), find a matrix of T.

Example 2.7.3Let R correspond to a right-hand rotation of a rigid body about thex3-axis by an angle teta. Find a matrix of R.

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Example 2.7.4Obtain the matrix for the tensor T, which transforms the base vectors as follows:

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Example 2.8.1Given that a tensor T transforms the base vectors as follows:

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Example 2.10.1(a) Let R correspond to a 90 right-hand rigid body rotation about the x3-axis. Find the matrix of R.(b) Let S correspond to a 90 right-hand rigid body rotation about the x1-axis. Find the matrix of S.(c) Find the matrix of the tensor that corresponds to the rotation R, followed by S.(d) Find the matrix of the tensor that corresponds to the rotation S, followed by R.(e) Consider a point P whose initial coordinates are (1,1,0). Find the new position of this point after the rotations of part (c). Also find the new position of thispoint after the rotations of part (d).

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In terms of tensor components

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Example 2.13.1Show that for any second-order tensor A and B

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Example 2.18.1

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Example 2.20.1Given

(a) Decompose the tensor into a symmetric and an antisymmetric part.(b) Find the dual vector for the antisymmetric part.

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Example 2.22.1Find the eigenvalues and eigenvectors for the tensor

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Example 2.22.2Find the eigenvalues and eigenvectors for the tensor

Example 2.22.3Find the eigenvalues and eigenvectors for the tensor

Example 2.22.4

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Example 2.22.4Find the eigenvalues and eigenvectors for the tensor

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Example 2.24.1For the tensor T, first find the principal scalar invariants and then evaluate the eigenvalues.

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