Page 87 - Introduction to Continuum Mechanics

72 Problems

f
(b) By using the vector transformation law, find the components of a = V 3e 1+e 2 in the primed
basis (i.e., find a/)

(c) Do part (b) geometrically.
2B18. Do the previous problem with e/ obtained by a 30° clockwise rotation of the e/-basis
about 63.
2B19. The matrix of a tensor T in respect to the basis {e/} is

Find TII, Ti2 and T^i in respect to a right-hand basis e/ where ej is in the direction of
-62+263 and 62 is in the direction of ej
2B20 (a) For the tensor of the previous problem, find [7/y] if e/ is obtained by a 90° right-hand
rotation about the C3-axis.
(b) Compare both the sum of the diagonal elements and the determinants of [T] and [T]'.

2B21. The dot product of two vectors a = a/e/ and b/ = ft/e/ is equal to a/6/. Show that the dot
product is a scalar invariant with respect to an orthogonal transformation of coordinates,
2B22. (a) If TIJ are the components of a tensor, show that T/jTJy is a scalar invariant with respect
to an orthogonal transformation of coordinates.
(b) Evaluate 7/,-T/,- if in respect to the basis e/

(d) Show for this specific [T] and [T]' that

2B23. Let [T] and [T]' be two matrices of the same tensor T, show that

2B24. (a) The components of a third-order tensor are R^. Show that/?//* are components of
a vector.