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2 a33 a31 al3 z i aii a12 i a32 a23 i aiia33 1 a21

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Unformatted text preview: I A.2 A33 A31 Al3 /z = I Aii A12 I + A32 A23/ + I AiiA33 1 A21 A22 13 = det(A¡j). 5 . c . CT = CT,.; C = 8, Ch 2: Problem 5 ker delta. nec where C is the direction cosine matrx and 8 is the matrx of the Kro ause it Any matrx obeying spch a relationship is called an orthogonal matri bec represents transformation of one set of ortogonal axes into another. 5. Show that for a second-order tensor A, the following thee quantities are invarant under the rotation of axes: II = A¡¡ I A.2 A33 A31 Al3 /z = I Aii A12 I + A32 A23/ + I AiiA33 1 A21 A22 Supplemental Readitig 13 = det(A¡j). show that (Hint; Use the result of Exercise 4 and the transformation rule (2.12) to s. In jkAki are also invarant i; = A;i = Aii = ii. Then show that AijAji and AijA fact, all contracted scalars of the form Aij A jk . . . Ami are invarian that ts. Finally, verify /z = l(ll - AijAji) h = AijAjkAki - !¡AijAji + /zAii. Because the right-hand sides are invarant, so are /z and h) ion 6. If u and v are vectors, show that the products Ui v j obey the transformat rule (2.1...
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