Page 176 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 15: Tensors in Cartesian Coordinates

Choosing another base e ,e ,e , and repeating this operation we get Notes
  
1
2
3
1 i ...i
Y  1 j ...j s r   X  1 i ...i r s . ...(2)
1 j ....j
Exercise 15.22: Prove that arrays Y 1 j ...j s r and Y 1 j ....j are related to each other in the same way as
1 i ...i
1 i ...i

r
s
arrays X 1 i ...i r s  X 1 i ...i r s , i.e. according to transformation formulas. In doing this you prove that

1 j ...j
1 j ....j
formula (1) applied in all bases produces new tensor Y = X from initial tensor X.
Formula (1) defines the multiplication of tensors by numbers. In exercise 15.22 you prove its
consistence. The next formula defines the addition of tensors:
X 1 i ...i r  Y 1 i ...i r  Z 1 i ...i r . ...(3)
1 j ...j s 1 j ....j s 1 j ...j s
Having two tensors X and Y both of type (r, s) we produce a third tensor Z of the same type (r, s)
by means of formula (3). It’s natural to denote Z = X + Y.
Exercise 2.1: By analogy with exercise 15.22 prove the consistence of formula (3).
Exercise 3.1: What happens if we multiply tensor X by the number zero and by the number
minus one? What would you call the resulting tensors?

15.8 Tensor Product

The tensor product is defined by a more tricky formula. Suppose we have tensor X of type (r, s)
and tensor Y of type (p, q), then we can write:

Z 1 i ...i r p  X 1 i ...i r s Y j  i  r 1 ...i s p  . ...(1)
r p

s 1 ...j
1 j ...j
1 j ....j
s p


Formula (1) produces new tensor Z of the type (r + p, s + q). It is called the tensor product of X and
Y and denoted Z = X  Y. Don’t mix the tensor product and the cross product. They are different.
Exercise 15.23: By analogy prove the consistence of formula (1).
Exercise 15.24: Give an example of two tensors such that X  Y  Y  X.
15.9 Contraction

As we have seen above, the tensor product increases the number of indices. Usually the tensor
Z = X  Y has more indices than X and Y. Contraction is an operation that decreases the number
of indices. Suppose we have tensor X of the type (r + 1, s + 1). Then we can produce tensor Z of
type (r, s) by means of the following formula:

n
Z 1 i ....i s  X 1 i ...i k 1 j ...j r . ...(1)
m 1 i ...i
m


r
1 j ...j
1 j ...j  
 1 k s
What we do ? Tensor X has at least one upper index and at least one lower index. We choose the
m-th upper index and replace it by the summation index . In the same way, we replace the k-th
lower index by . Other r upper indices and s lower indices are free. They are numerated in some
convenient way, say as in formula (1). Then we perform summation with respect to index . The
contraction is over. This operation is called a contraction with respect to m-th upper and
k-th lower indices. Thus, if we have many upper an many lower indices in tensor X, we can
perform various types of contractions to this tensor.

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