Page 26 - Introduction to Continuum Mechanics

Part B Tensor - A Linear Transformation 11

Part B Tensors

281 Tensor - A Linear Transformation
Let T be a transformation, which transforms any vector into another vector. If T transforms
a into c and b into d, we write Ta = c and Tb = d.
If T has the following linear properties:

where a and b are two arbitrary vectors and a is an arbitrary scalar then T is called a linear
transformation. It is also called a second-order tensor or simply a tensor. An alternative and
equivalent definition of a linear transformation is given by the single linear property:

where a and b are two arbitrary vectors and a and ft are arbitrary scalars.
If two tensors, T and S, transform any arbitrary vector a in an identical way, then these
tensors are equal to each other, i.e., Ta=Sa -» T=S.

Let T be a transformation which transforms every vector into a fixed vector n. Is this
transformation a tensor?
Solution. Let a and b be any two vectors, then by the definition of T,
Ta = n, Tb = n and T(a+b) = n
Clearly,
T(a+b) * Ta+Tb
Thus, T is not a linear transformation. In other words, it is not a tensor.

t Scalars and vectors are sometimes called the zeroth and first order tensor, respectively. Even though they can
also be defined algebraically, in terms of certain operational rules, we choose not to do so. The geometrical
concept of scalars and vectors, which we assume that the students are familiar with, is quite sufficient for our
purpose.