Page 84 - Introduction to Continuum Mechanics

Tensors 69

(a) Evaluate [7^] if T fj = £ ijl(a k
(b) Evaluate [qj if q = £iji<Sjk
(c) Evaluate [di\ if 4t = £pa,-6/ and show that this result is the same as d* = (axb) • %

2A6.
(a) If BifiTjk = 0,show that 7)y = 7},-
(b) Show that 6^^ - 0
2A7. (a)Verify that

E e
ijm klm ~ &a£jl'~&ifijk
By contracting the result of part (a) show that
e
(b)%m //m = 2*5,y
=
(fyij&ijk 6
2A8. Using the relation of Problem 2A7a, show that
ax(bxc) = (a-c)b-(a-b)c
2A9. (a) If TII = - Tji show that TqagOj = 0
(b) If T fj = -T^ and 5,y = Sj it show that T MS kt = 0

2A10. Let 7« = -(Sjj+Sji) and /?« = -(Sy—5 ;-j), show that
4f~r <w
=
% Tjj+Rij, TJJ = 7J/, and /?,y = -/?y/
x
2A11. Let f(xi,X2f 3) be a function of jc,- and v/fo^^a) represent three functions of jc,-. By
expanding the following equations, show that they correspond to the usual formulas of
f-\*-TT^S«*^kft4-1 o I y^rk t^»i 11 if

2A12. Let |/4,y| denote the determinant of the matrix [yi^]. Show that |y4,y| — ^ijk^iV^j2^k3'
n
2B1. A transformation T operates on a vector a to give Ta = T—r, where j a | is the magnitude
a
l l
of a. Show that T is not a linear transformation.
2B2. (a) A tensor T transforms every vector a into a vector Ta = m x a, where m is a specified
vector. Prove that T is a linear transformation.
(b) If m = ej + 62, find the matrix of the tensor T