7/29/2019 The summation convention

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The summation convention

In this chapter we have often needed to take a sum over a number of terms which are allof the same general form, and differ only in the value of an indexing subscript. Such a

summation has been indicated by a summation sign,

_

, with the range of the subscriptwritten above and below the sign. This very explicit notation has been deliberately adopted

for the purposes of introducing the general procedures. However, the reader will, aftera time, doubtless have felt that much of the notation is superfluous, particularly when

there have been multiple sums appearing in a single expression, each with its own explicit

summation sign; the derivation of equation (1.99) provides just such an example.

Such calculations can be significantly compacted, and in some cases simplified, if the

Cartesian coordinates x, yand zare replaced symbolically by the indexed coordinates xi,

where itakes the values 1, 2 and 3, and the so-calledsummation convention is adopted. In

this convention any lower-case alphabetic subscript that appears exactly twice in any term

of an expression is understood to be summed over all the values that a subscript in thatposition can take (unless the contrary is specifically stated); there is no explicit summation

sign.

The subscripted quantities may appear in the numerator and/or the denominator of aterm in an expression. This naturally implies that any such pair of repeated subscripts

must occur only in subscript positions that have the same range of values. Sometimes theranges of values have to be specified, but usually they are apparent from the context.

As a basic example, in this notation

Pij=

_N

k=1

AikBkj

becomes

Pij= AikBkji.e. without the explicit summation sign.

In order to use the convention, partial differentiation with respect to Cartesian coordinatesx, yand zis denoted by the generic symbol/xi; this facilitates a compact

and efficient notation for the development of vector calculus identities. These are studied

in Chapter2, though, for the same reasons that matrix algebra was first presented herewithout using the convention, vector calculus is initially developed there without recourse

to it.

Further discussion of the summation convention, together with additional examples

of it use, form the content of Appendix D. Considerable care is needed when using the

convention, but mastering it is well worthwhile, as it considerably shortens many matrixalgebra and vector calculus calculations.