1. NPTEL β Physics β Mathematical Physics - 1
Lecture 28
Algelraic properties of tensors
Like vectors, the tensors obey certain operations which are:
a) Addition
If T and S are two tensors of type (r, s), then their sum U = T + S is defined as
ππ1β¦β¦β¦β¦β¦β¦
π
π
π1β¦β¦β¦β¦..ππ
= ππ1β¦β¦..ππ
+ ππ1β¦β¦..π
π
π 1β¦β¦β¦π π π 1β¦β¦β¦π
π
Thus U is also a tensor of type (r, s), which can be easily proved by showing the
transformation property,
πΜ π1β¦β¦β¦β¦..ππ
=
ππ₯Μ
π 1β¦β¦β¦β¦β¦β¦π
π
ππ₯β1
π1 π
π
ππ₯Μ ππ₯
β¦ β¦ β¦
ππ₯βπ ππ₯Μ π1
β¦ β¦ β¦
ππ₯Μ ππ
π1
ππ₯ππ
πβ1β¦β¦β¦β¦..βπ
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π1β¦β¦β¦β¦β¦β¦π
π
b) Multiplication
If T is a tensor of type (π1, π 1) and S is a tensor of type (π2, π 2),
then the product U = T β S, defined component wise as,
ππ 1β¦β¦β¦β¦β¦β¦
π
π 1+π 2 π 1 1 π 1+π 2
π1β¦β¦β¦β¦..ππ 1+π 2
= π
π1β¦β¦β¦β¦..ππ 1
π
πππ 1+1 β¦β¦β¦β¦..ππ 1+
π 2 π 1β¦β¦β¦β¦β¦β¦π π π +1β¦β¦β¦β¦β¦β¦
π
Which is a tensor of type (π1 + π2, π 1 + π 2). For example, if T is a tensor of type
π ππ
(1,2) with components ππ π and S is a tensor type (2,1) with components ππ
,
then the components of the tensor product U as,
ππππ ππ π
and they transform according to the rules,
πππ = ππ ππ
π
π
Μ π ππ π π
π
ππ
π
= π π =
Μ Μ
π ππ π
ππ₯
ππ‘
ππ
The above shows that U is a tensor of rank (3, 3).
c) Contraction
The contraction is defined by the following operation β given by a tensor
type (r, s), take a covariant index and set it equal to a contravariant index, that
is, sum over those two indices. It will result in a tensor of type (r-1, s-1). An
example will make it clear. Take a tensor of type (2, 1) whose
π₯Μ π ππ₯ π ππ₯2 ππ₯Μ π ππ₯Μ π ππ₯π‘
π
β ππ₯Μ π
π
π₯Μ 2 πππ ππ₯π ππ₯π π π₯π
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NPTEL β Physics β Mathematical Physics - 1
components are ππ π
and set k = j. Now how do the components of ππ
π
π π
transform?
π =
Μ π
π
π
ππ₯Μ ππ₯Μ ππ₯ βπ
=
ππ₯Μ
π π π π
ππ₯π ππ₯π ππ₯Μ π ππ
ππ₯β
πβπ
π
This shows that ππ transforms as components of a contravariant tensor of type
(1, 0).
Of specific interest is a tensor of type (1, 1). Contracting this, one will get a
tensor of type (0, 0) i.e. a scalar. Let π΄β is a contravariant vector with
components π΄π and π΅ββ is a covariant vector with components π΅π. Then
ππ = π΄ π΅π is a tensor of type (1, 1). When one contracts it, one gets ππ = A Bi
which is a scalar as we have taken dot product of two vectors.
π
π
π π π i
Symmetrization
Some of the tensors we come across in physics have the property that when
two of their indices are interchanged, the tensors either change or do not
change sign. The ones which do not change sign are called as
symmetric tensors and those which change sign under change of indices
are called as antisymmetric tensors. Examples are β
πππ = πji ; π is a symmetric tensor
πππ = βπji ; π is an antisymmetric tensor