Page 57 - Intro to Tensor Calculus

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Contraction

The operation of contraction on any mixed tensor of rank m is performed when an upper index is
set equal to a lower index and the summation convention is invoked. When the summation is performed
over the repeated indices the resulting quantity is also a tensor of rank or order (m − 2). For example, let
i
A , i,j,k =1, 2,...,N denote a mixed tensor and perform a contraction by setting j equal to i. We obtain
jk
A i = A 1 + A 2 + ··· + A N (1.2.57)
ik 1k 2k Nk = A k
i
where k is a free index. To show that A k is a tensor, we let A ik = A k denote the contraction on the
i
transformed components of A . By hypothesis A i is a mixed tensor and hence the components must
jk jk
satisfy the transformation law
n
i
i m ∂x ∂x ∂x p
A jk = A np .
j
∂x m ∂x ∂x k
Now execute a contraction by setting j equal to i and perform a summation over the repeated index. We
find
i
n
n
i m ∂x ∂x ∂x p m ∂x ∂x p
A ik = A k = A np i k = A np k
∂x m ∂x ∂x ∂x ∂x (1.2.58)
m
∂x p n ∂x p ∂x p
m
n
= A δ = A np = A p .
np m
∂x k ∂x k ∂x k
Hence, the contraction produces a tensor of rank two less than the original tensor. Contractions on other
mixed tensors can be analyzed in a similar manner.
New tensors can be constructed from old tensors by performing a contraction on an upper and lower
index. This process can be repeated as long as there is an upper and lower index upon which to perform the
contraction. Each time a contraction is performed the rank of the resulting tensor is two less than the rank
of the original tensor.
Multiplication (Inner Product)
The inner product of two tensors is obtained by:
(i) first taking the outer product of the given tensors and
(ii) performing a contraction on two of the indices.
EXAMPLE 1.2-5. (Inner product)
i
Let A and B j denote the components of two first order tensors (vectors). The outer product of these
tensors is
i
i
C = A B j ,i, j =1, 2,... ,N.
j
The inner product of these tensors is the scalar
2
1
N
i
C = A B i = A B 1 + A B 2 + ··· + A B N .
Note that in some situations the inner product is performed by employing only subscript indices. For
example, the above inner product is sometimes expressed as
C = A i B i = A 1 B 1 + A 2 B 2 + ··· A N B N .

This notation is discussed later when Cartesian tensors are considered.