Introduction to vector and tensor analysis

n.
f = σ t · n
A tensor often associated with the stress tensor is the strain tensor, U, that
specifies how a strained material is distorted ∆u in some direction ∆r
∆u = U · ∆r
We shall be more specific on these physical tensors later.
3.2 Outer product
The outer product a ⊗ b or simply ab between two vectors a and b is a tensor
defined by the following equation, where c is any vector
(ab)(c) = a(b · c) (3.4)
In words, for each c the tensor ab associates a vector in the direction of a and
with a magnitude equal to the projection of c into b. In order to call the object
(ab) a tensor we should verify that it is a linear operator. Using the definition,
Eq. (3.3), allows us to “place the brackets where we want”
(ab) · c = a(b · c),
which will ease the notation. Now, demonstrating that (ab) indeed is a a tensor
amounts to “moving brackets”:
(ab) · (mc) = a(b · (mc)) = ma(b · c)
= m(ab) · c
(ab) · (c + d) = a(b · (c + d)) = a(b · c) + a(b · d)
= (ab) · c + (ab) · d
(3.5)
We note that since the definition, eq. (??) involves vector operations that
bear no reference to coordinates/components, the outer product will itself be
invariant to the choise of coordinate system. The outer product is also known
in the litterature as the tensor, direct, exterior or dyadic product. The tensor
formed by the outer product of two vectors is called a dyad.
3.3 Basic tensor algebra
We can form more general linear vector operators by taking sum of dyads. The
sum of any two tensors,S and T, and the multiplication of a tensor with a scalar
m are naturally defined by
(S + T) · c = S · c + T · c
(mS) · c = m(S · c).
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(3.6)