Manipulations with the Indicial Notations

(a) Substitution
If

$$a_i = v_{im}b_m$$ (2.6.1)

and

$$b_i = v_{im}c_m,$$ (2.6.2)

then, in order to substitute for $b_i$'s from (2.6.2) into (2.6.1) we first change the dummy index from $m$ to some other letter, say $n$ and then the free index in (2.6.2) from $i$ to $m$, so that

$$b_m = v_{mn}c_n.$$ (2.6.3)

Now (2.6.1) and (2.6.3) give

$$a_i = v_{im}v_{mn}c_n.$$ (2.6.4)

Note that (2.6.4) represents three equations each having the sum of nine terms on its right-hand side.

(b) Multiplication

$$\noalign{\mbox{If}} p = a_mb_m$$ (2.6.5)

$$\noalign{\mbox{and}} q = c_md_m$$ (2.6.6)

$$\noalign{\mbox{then}} pq = a_mb_mc_nd_n.$$ (2.6.7)

It is important to note that $pq\ne a_mb_mc_md_m$. In fact the right-hand side of this expression is not even defined in the summation convention and further it is obvious that

$$pq\ne \sum^3_{m=1} a_mb_mc_md_m.$$

(c) Factoring
If

$$T_{ij}n_j - \lambda n_i = 0,$$ (2.6.8)

then, using the Kronecker delta, we can write

$$n_i = \delta_{ij}n_j,$$ (2.6.9)

so that (2.6.8) becomes

$$T_{ij}n_j - \lambda \delta_{ij}n_j = 0.$$

Thus

$$(T_{ij} - \lambda\delta_{ij})n_j = 0.$$

(d) Contraction

The operation of identifying two indices and so summing on them is known as contraction. For example, $T_{ii}$ is the contraction of $T_{ij}$,

$$T_{ii} = T_{11} + T_{22} + T_{33},$$

and $T_{ijj}$ is a contraction of $T_{ijk}$,

$$T_{ijj} = T_{i11}+ T_{i22} + T_{i33}.$$

If

$$T_{ij} = \lambda E_{kk}\delta_{ij} + 2\mu E_{ij},$$

then

$$T_{ii} =\ \lambda E_{kk}\delta_{ii} + 2\mu E_{ii},$$

$$=\ (3\lambda + 2\mu )E_{kk}.$$

Exercise. Given that $T_{ij} = \lambda E_{kk}\delta_{ij} + 2\mu E_{ij}$, $W = {\scriptstyle \frac{1}{2}} T_{ij}E_{ij},\ P = T_{ij}T_{ij}$, show that

$$W =\ \mu E_{ij}E_{ij} + \frac{\lambda}{2}(E_{kk})^2,$$

$$P =\ 4\mu^2E_{ij}E_{ij} + (E_{kk})^2(4\mu\lambda + 3\lambda^2).$$