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- Question : 1E - For the given matrix/vector pairs, compute the following quantities: aii, aijaij, aijajk, aijbj, aijbibj, bibj, bibi. 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 aijajk is the product [a][a].
- Question : 2E - Use the decomposition result (1.2.10) to express aij from Exercise 1.1 in terms of the sum of symmetric and antisymmetric matrices. Verify that a(ij) and a[ij] satisfy the conditions given in the last paragraph of Section 1.2.
- Question : 3E - If aij is symmetric and bij is antisymmetric, prove in general that the product aijbij is zero. Verify this result for the specific case by using the symmetric and antisymmetric terms from Exercise 1.2.
- Question : 4E - Explicitly verify the following properties of the Kronecker delta dijaj
- Question : 5E - Formally expand the expression (1.3.4) for the determinant and justify that either index notation form yields a result that matches the traditional form for det[aij].
- Question : 6E - 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 (p/4 radians) about the x1-axis. The rotation direction follows the positive sense presented in Example 1.2.
- Question : 7E - Consider the two-dimensional coordinate transformation shown in Figure 1.7. Through the counterclockwise rotation q, a new polar coordinate system is created. Show that the transformation matrix for this case is given by Qij
- Question : 8E - Show that the second-order tensor adij, where a is an arbitrary constant, retains its form under any transformation Qij. This form is then an isotropic second-order tensor.
- Question : 9E - The most general form of a fourth-order isotropic tensor can be expressed by adijdkl
- Question : 10E - For the fourth-order isotropic tensor given in Exercise 1.9, show that if b
- Question : 11E - Show that the fundamental invariants can be expressed in terms of the principal values as given by relations (1.6.5).
- Question : 12E - Determine the invariants, and principal values and directions of the following matrices. Use the determined principal directions to establish a principal coordinate system, and following the procedures in Example 1.3, formally transform (rotate) the given matrix into the principal system to arrive at the appropriate diagonal form.
- Question : 13E - A second-order symmetric tensor field is given by aij
- Question : 14E - Calculate the quantities V$u; V u; V2u; Vu; tr
- Question : 15E - The dual vector ai of an antisymmetric second-order tensor aij is defined by ai
- Question : 16E - Using index notation, explicitly verify the vector identities: (a) (1.8.5)1,2,3 (b) (1.8.5)4,5,6,7 (c) (1.8.5)8,9,10
- Question : 17E - Extend the results found in Example 1.5, and determine the forms of Vf ; V$u;V2f ; and V u for a three-dimensional cylindrical coordinate system (see Figure 1.5).
- Question : 18E - For the spherical coordinate system (R, f, q) in Figure 1.6, show that h1

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