2.2 Vectors
21
To save time and effort, Einstein (1916) developed a shorthand notation for tensor
analysis. According to the summation convention, if an index appears twice in a
single term, summation is implied over the range for that index. 5 Accordingly, we
can write the expression
3
i=1 a i b i in the abbreviated form a i b i without needing
to explicitly write the summation symbol, as summation over the repeated index i
is implied. In contrast, summation over i is not implied in the expression a i + b i ,
because a i and b i are two separate terms.
Note that the meaning of a i b i is independent of the specific symbol used for
the subscript, i.e., a i b i = a j b j = a 1 b 1 + a 2 b 2 + a 3 b 3 after the summation over
i or j is carried out. For this reason, the repeated index i (or j ) is called a dummy
index, because it can be replaced by any letter without altering the meaning of an
expression. As we will see, changing the symbols used in dummy indices can be an
extremely useful and sometimes necessary operation.
In this and the next chapter, the summation convention is assumed to hold unless
stated otherwise or if subscripts are enclosed in parentheses, e.g., writing a (i) b (i) or
a (ii) indicates that summation over i is not implied. Furthermore, an index that is
not repeated (a free index) implies a set of equations. For example,
a i b ij = a 1 b 1j + a 2 b 2j + a 3 b 3j = 0
represents three different equations, one for each j = 1, 2, 3, i.e.,
a 1 b 11 + a 2 b 21 + a 3 b 31 = 0
a 1 b 12 + a 2 b 22 + a 3 b 32 = 0
a 1 b 13 + a 2 b 23 + a 3 b 33 = 0.
Like units, the indices in an equation must “balance” in that all terms must
contain the same free indices. Thus, a i + b ij c j = 0 is a valid equation, since
both terms contain an i after summing over j , but a ik + b ij c j = 0 is not valid,
because the first term contains an extra k. Moreover, unless some indices are placed
in parentheses, writing three or more of the same index in a single term is not
permitted.
2.2 Vectors
Before delving into second-order tensors, we consider some basic concepts in vector
mechanics.
5 Throughout this book, the summation convention applies only to the indices i, j, k, l, m, n with
the default range being 1, 2, 3 (in 3D space). Thus, summation over x is not implied for a xx .
21
To save time and effort, Einstein (1916) developed a shorthand notation for tensor
analysis. According to the summation convention, if an index appears twice in a
single term, summation is implied over the range for that index. 5 Accordingly, we
can write the expression
3
i=1 a i b i in the abbreviated form a i b i without needing
to explicitly write the summation symbol, as summation over the repeated index i
is implied. In contrast, summation over i is not implied in the expression a i + b i ,
because a i and b i are two separate terms.
Note that the meaning of a i b i is independent of the specific symbol used for
the subscript, i.e., a i b i = a j b j = a 1 b 1 + a 2 b 2 + a 3 b 3 after the summation over
i or j is carried out. For this reason, the repeated index i (or j ) is called a dummy
index, because it can be replaced by any letter without altering the meaning of an
expression. As we will see, changing the symbols used in dummy indices can be an
extremely useful and sometimes necessary operation.
In this and the next chapter, the summation convention is assumed to hold unless
stated otherwise or if subscripts are enclosed in parentheses, e.g., writing a (i) b (i) or
a (ii) indicates that summation over i is not implied. Furthermore, an index that is
not repeated (a free index) implies a set of equations. For example,
a i b ij = a 1 b 1j + a 2 b 2j + a 3 b 3j = 0
represents three different equations, one for each j = 1, 2, 3, i.e.,
a 1 b 11 + a 2 b 21 + a 3 b 31 = 0
a 1 b 12 + a 2 b 22 + a 3 b 32 = 0
a 1 b 13 + a 2 b 23 + a 3 b 33 = 0.
Like units, the indices in an equation must “balance” in that all terms must
contain the same free indices. Thus, a i + b ij c j = 0 is a valid equation, since
both terms contain an i after summing over j , but a ik + b ij c j = 0 is not valid,
because the first term contains an extra k. Moreover, unless some indices are placed
in parentheses, writing three or more of the same index in a single term is not
permitted.
2.2 Vectors
Before delving into second-order tensors, we consider some basic concepts in vector
mechanics.
5 Throughout this book, the summation convention applies only to the indices i, j, k, l, m, n with
the default range being 1, 2, 3 (in 3D space). Thus, summation over x is not implied for a xx .