Page 160 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 14: Notation and Summation Convention

It is convenient to use a summation convention for repeated letter suffixes. According to this Notes
convention, if a letter suffix occurs twice in the same term, a summation over the repeated suffix
from 1 to 3 is implied without a summation sign, unless otherwise indicated. For example,
equation (1) can be written as
u = u e = u e + u e + u e ...(2)
3 3
2 2
i i
1 1
without the summation sign. The sum of two vectors is commutative and is given by
u + v = v + u = (u + v )e , i
i
i
which is consistent with the parallelogram rule. A further example of the summation convention
is the scalar or inner product of two vectors,
u . v = u v = u v + u v + u v ...(3)
2 2
1 1
3 3
i i
Repeated suffixes are often called dummy suffixes since any letter that does not appear elsewhere
in the expression may be used, for example,
u v = uv.
i i
j j
Equation (3) indicates that the scalar product obeys the commutative law of algebra, that is,
u . v = v . u.
The magnitude |u| of a vector u is given by

|u| u.u  u u .
i
i
Other examples of the use of suffix notation and the summation convention are
C = C + C + C 33
22
11
ii
C b = C b + C b + C b .
i2 2
ij j
i3 3
i1 1
A suffix that appears once in a term is known as a free suffix and is understood to take in turn the
values 1, 2, 3 unless otherwise indicated. If a free suffix appears in any term of an equation or
expression, it must appear in all the terms.
14.3 Orthogonal Transformations
The scalar products of orthogonal unit base vectors are given by

e . e =  , ...(1)
j
i
ij
where  is known as the Kronecker delta and is defined as
ij
 1 for i  j
   . ...(2)
ij
 0 for i  j
The base vectors ei are orthonormal, that is, of unit magnitude and mutually perpendicular to
each other. The Kronecker delta is sometimes called the substitution operator because
u + u  + u  + u  = u . i ...(3)
1 i1
j ij
2 i2
3 i3
Consider a right-handed rectangular Cartesian coordinate system 0x’ with the same origin as
i
0x as indicated in Figure 14.2. Henceforth, primed quantities are referred to coordinate system
i
0x’. i

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