Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 368

or A T w ¼ q Det A and since w ¼ ( A b A c ), q ¼ ( b c ),

A T ð A b A c Þ¼ð b c Þ Det A ; or

Æ

ð A b A c Þ A ¼ð b c Þ Det A ;

Problems

A.8.1. Find the cross products a b and b a of the two vectors a ¼ [1, 2, 3]

and b ¼ [4, 5, 6]. What is the relationship between a b and b a ?

A T , then A.8.2. Show that if A is a skew-symmetric 3 by 3 matrix, A ¼

Det A

¼

0.
A.8.3. Evaluate Det( a

b ).

Det A T . A.8.4. Show Det A ¼

1if Q T Q T A.8.5. Show Det Q 1 .

A.8.6. Find the volume of the parallelepiped if the three nonparallel edges of a
parallelepiped coincide with the three vectors a , b , and c where a

¼ Q ¼ Q ¼

¼ [1, 2, 3]

meters, b ¼ [1, 4, 6] meters and c ¼ [1, 1, 1] meters.

A.8.7. If v x and a is a constant vector, using the indicial notation, evaluate

the div v and the curl v .

¼ a

A.9 The Moment of Inertia Tensor

The mass moment of inertia tensor illustrates many features of the previous sections
such as the tensor concept and definition, the open product of vectors, the use of unit
vectors and the significance of eigenvalues and eigenvectors. The mass moment of
inertia is second moment of mass with respect to an axis. The first and zeroth
moment of mass with respect to an axis is associated with the concepts of the center
of mass of the object and the mass of the object, respectively. Let d v represent the
differential volume of an object O . The volume of that object V O is then given by

ð

V O ¼ d v ; (A.118)

O

and, if r ( x 1 , x 2 , x 3 , t ) ¼ r ( x , t ) is the density of the object O , then the mass M O of O
is given by

ð
O rðx;

M O ¼ t Þ d v : (A.119)

The centroid x centroid and the center of mass x cm of the object O are defined by

ð ð

1
V O

1
M O

x centroid ¼ x d v ; x cm ¼ x rð x ; t Þ d v (A.120)

O O