Page 6 - Intro to Tensor Calculus

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It is also convenient at this time to mention that higher dimensional vectors may be defined as ordered
n−tuples. For example, the vector
~
X =(X 1 ,X 2 ,...,X N )
with components X i ,i =1, 2,... ,N is called a N−dimensional vector. Another notation used to represent
this vector is
~
X = X 1 b e 1 + X 2 b e 2 + ··· + X N b e N
where

b e 1 , b e 2 ,... , b e N
are linearly independent unit base vectors. Note that many of the operations that occur in the use of the
index notation apply not only for three dimensional vectors, but also for N−dimensional vectors.
In future sections it is necessary to define quantities which can be represented by a letter with subscripts
or superscripts attached. Such quantities are referred to as systems. When these quantities obey certain
transformation laws they are referred to as tensor systems. For example, quantities like

A k e ijk δ j A i a ij .
ij δ ij i B j

The subscripts or superscripts are referred to as indices or suffixes. When such quantities arise, the indices
must conform to the following rules:
1. They are lower case Latin or Greek letters.
2. The letters at the end of the alphabet (u, v, w, x, y, z) are never employed as indices.

The number of subscripts and superscripts determines the order of the system. A system with one index
is a first order system. A system with two indices is called a second order system. In general, a system with
N indices is called a Nth order system. A system with no indices is called a scalar or zeroth order system.
The type of system depends upon the number of subscripts or superscripts occurring in an expression.
m
For example, A i and B , (all indices range 1 to N), are of the same type because they have the same
jk st
number of subscripts and superscripts. In contrast, the systems A i and C mn are not of the same type
jk p
because one system has two superscripts and the other system has only one superscript. For certain systems
the number of subscripts and superscripts is important. In other systems it is not of importance. The
meaning and importance attached to sub- and superscripts will be addressed later in this section.
In the use of superscripts one must not confuse “powers ”of a quantity with the superscripts. For
1
2
3
example, if we replace the independent variables (x, y, z)by the symbols (x ,x ,x ), then we are letting
3
2
2
y = x where x is a variable and not x raised to a power. Similarly, the substitution z = x is the
3
replacement of z by the variable x and this should not be confused with x raised to a power. In order to
2 3
2
write a superscript quantity to a power, use parentheses. For example, (x ) is the variable x cubed. One
of the reasons for introducing the superscript variables is that many equations of mathematics and physics
can be made to take on a concise and compact form.
There is a range convention associated with the indices. This convention states that whenever there
is an expression where the indices occur unrepeated it is to be understood that each of the subscripts or
superscripts can take on any of the integer values 1, 2,...,N where N is a specified integer. For example,