Page 8 - Intro to Tensor Calculus

4

Summation Convention

The summation convention states that whenever there arises an expression where there is an index which
occurs twice on the same side of any equation, or term within an equation, it is understood to represent a
summation on these repeated indices. The summation being over the integer values specified by the range. A
repeated index is called a summation index, while an unrepeated index is called a free index. The summation
convention requires that one must never allow a summation index to appear more than twice in any given
expression. Because of this rule it is sometimes necessary to replace one dummy summation symbol by
some other dummy symbol in order to avoid having three or more indices occurring on the same side of
the equation. The index notation is a very powerful notation and can be used to concisely represent many
complex equations. For the remainder of this section there is presented additional definitions and examples
to illustrated the power of the indicial notation. This notation is then employed to define tensor components
and associated operations with tensors.

EXAMPLE 1.1-1 The two equations

y 1 = a 11 x 1 + a 12 x 2
y 2 = a 21 x 1 + a 22 x 2

can be represented as one equation by introducing a dummy index, say k, and expressing the above equations
as
y k = a k1 x 1 + a k2 x 2 , k =1, 2.

The range convention states that k is free to have any one of the values 1 or 2, (k is a free index). This
equation can now be written in the form
2
X
y k = a ki x i = a k1 x 1 + a k2 x 2
i=1
where i is the dummy summation index. When the summation sign is removed and the summation convention
is adopted we have
i, k =1, 2.
y k = a ki x i
Since the subscript i repeats itself, the summation convention requires that a summation be performed by
letting the summation subscript take on the values specified by the range and then summing the results.
The index k which appears only once on the left and only once on the right hand side of the equation is
called a free index. It should be noted that both k and i are dummy subscripts and can be replaced by other
letters. For example, we can write
n, m =1, 2
y n = a nm x m
where m is the summation index and n is the free index. Summing on m produces

y n = a n1 x 1 + a n2 x 2

and letting the free index n take on the values of 1 and 2 we produce the original two equations.