Introduction to vector and tensor analysis

Rank 2 Tensors
A rank two tensor is a 3 × 3 matrix of real numbers, T , which transforms as the
outer product of two vectors
T ′
ij = aikajlTkl
Again it should be noticed that it is the transformation rule which guarantees
that the nine quantities collected in a matrix form a tensor. The matrix notation
of this transformation is
T ′ = A · T · A t
4.3 Tensors of any rank
4.3.1 Introduction
Tensors of rank higher than two are obtained by a straightforward iteration of
the definition of the outer product. For instance the outer product between a
dyad a1a2 and a vector a3 can be defined as the linear operator, a1a2a3, that
maps any vector c into a tensor according to the rule
a1a2a3(c) = def. a1a2(a3 · c) (4.18)
In words, for each c the object a1a2a3 associates a tensor obtained by multiplying
the dyad a1a2 with the scalar a3 · c. Definition Eq. (4.18) preserves the
invariance with respect to the choise of coordinates. Addition of triple products
and scalar multiplication is defined analogous to the algebra for dyads.
In section 3.4 we demonstated that a basis for second order tensors is obtained
by forming all 3 × 3 dyad combinations, eiej, between orthonormal basis
vectors. Similarly, one can demonstrate that a basis for all linear operators,
T3, which map a vector into a second rank tensor is obtained by forming the
3 3 combinations of the direct products eiejek. Consequently, for any T3 there
exists a unique set of quantities, Tijk, such that
T3 = Tijkeiejek
NB! Einstein summation convention
Not surprisingly, these quantities, Tijk, are called the components of T3 and
since three indicies are needed, T3 is called a third rank tensor. Upon a coordinate
transformation we would need to transform the components of each
individual vector in the triple product. Expressing the original basis vectors in
terms of the new basis
one obtains
T3 = T ′
lmne′ le′ me′ n
ei = ajie ′ j
= Tijkeiejek
= Tijk(alie ′ l)(amje ′ m)(anke ′ n) = aliamjankTijke ′ le′ me ′ n
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(4.19)