General Mechanics/Index Notation

Summation convention

If we label the axes as 1,2, and 3 we can write the dot product as a sum

$𝐮\cdot 𝐯=\sum _{i=1}^3{u}_i{v}_i$

If we number the elements of a matrix similarly,

$𝐀=\left(\begin{array}{ccc}{A}_{11}& A_{12}& A_{13}\\ A_{21}& A_{22}& A_{23}\\ A_{31}& A_{32}& A_{33}\end{array}\right)𝐁=\left(\begin{array}{ccc}{B}_{11}& B_{12}& B_{13}\\ B_{21}& B_{22}& B_{23}\\ B_{31}& B_{32}& B_{33}\end{array}\right)$

we can write similar expressions for matrix multiplications

$(𝐀𝐮{)}_i=\sum _{j=1}^3{A}_{ij}{u}_j(𝐀𝐁{)}_{ik}=\sum _{j=1}^3{A}_{ij}{B}_{jk}$

Notice that in each case we are summing over the repeated index. Since this is so common, it is now conventional to omit the summation sign.

Instead we simply write

$𝐮\cdot 𝐯=u_i{v}_i(𝐀𝐮{)}_i=A_{ij}{u}_j(𝐀𝐁{)}_{ik}=A_{ij}{B}_{jk}$

We can then also number the unit vectors, ê_i, and write

$𝐮=u_i{\widehat{𝐞}}_i$

which can be convenient in a rotating coordinate system.

Kronecker delta

The Kronecker delta is

${\delta }_{ij}=\{\begin{array}{cc}1& i=j\\ 0& i\ne j\end{array}$

This is the standard way of writing the identity matrix.

Levi-Civita (Alternating) symbol

Another useful quantity can be defined by

${ϵ}_{ijk}=\{\begin{array}{cc}1& (i,j,k)=(1,2,3)\text{ or }(2,3,1)\text{ or }(3,1,2)\\ -1& (i,j,k)=(2,1,3)\text{ or }(3,2,1)\text{ or }(1,3,2)\\ 0& \text{ otherwise }\end{array}$

With this definition it turns out that

$𝐮\times 𝐯={ϵ}_{ijk}{\widehat{𝐞}}_i{u}_j{v}_k$

and

${ϵ}_{ijk}{ϵ}_{ipq}={\delta }_{jp}{\delta }_{kq}-{\delta }_{jq}{\delta }_{kp}$

This will let us write many formulae more compactly.