Determine the components of the vector bi and matrix aij given in Exercise 1.1 in a new coordinate system found through a rotation of 45° (π/4 radians) about the x1-axis. The rotation direction follows the positive sense presented in Example 1.2.

Data from exercise 1.1

For the given matrix/vector pairs, compute the following quantities: a_ii, a_ija_ij, a_ija_jk, a_ijb_j, a_ijb_ib_j, b_ib_j, b_ib_i. For each case, point out whether the result is a scalar, vector or matrix. Note that aijbj is actually the matrix product [a]{b}, while a_ija_jk is the product [a][a].

Example 1.2

Fig 1.2

Equation 1.5.1

Transcribed Image Text:

The components of a first- and second-order tensor in a particular coordinate frame are given by [1 0 37 022 324 a = H = Determine the components of each tensor in a new coordinate system found through a rotation of 60 (/3 radians) about the x3-axis. Choose a counterclockwise rotation when viewing down the negative x3-axis (see Fig. 1.2).