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

$\mathbf {u} \cdot \mathbf {v} =\sum _{i=1}^{3}u_{i}v_{i}$

If we number the elements of a matrix similarly,

$\mathbf {A} ={\begin{pmatrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{pmatrix}}\quad \mathbf {B} ={\begin{pmatrix}B_{11}&B_{12}&B_{13}\\B_{21}&B_{22}&B_{23}\\B_{31}&B_{32}&B_{33}\end{pmatrix}}$

we can write similar expressions for matrix multiplications

$(\mathbf {A} \mathbf {u} )_{i}=\sum _{j=1}^{3}A_{ij}u_{j}\quad (\mathbf {A} \mathbf {B} )_{ik}=\sum _{j=1}^{3}A_{ij}B_{jk}$

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

$\mathbf {u} \cdot \mathbf {v} =u_{i}v_{i}\quad (\mathbf {A} \mathbf {u} )_{i}=A_{ij}u_{j}\quad (\mathbf {A} \mathbf {B} )_{ik}=A_{ij}B_{jk}$

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

$\mathbf {u} =u_{i}{\hat {\mathbf {e} }}_{i}$

which can be convenient in a rotating coordinate system.

Kronecker delta

The Kronecker delta is

$\delta _{ij}=\left\{{\begin{matrix}1&i=j\\0&i\neq j\end{matrix}}\right.$

This is the standard way of writing the identity matrix.

Levi-Civita (Alternating) symbol

Another useful quantity can be defined by

$\epsilon _{ijk}=\left\{{\begin{matrix}1&(i,j,k)=(1,2,3){\mbox{ or }}(2,3,1){\mbox{ or }}(3,1,2)\\-1&(i,j,k)=(2,1,3){\mbox{ or }}(3,2,1){\mbox{ or }}(1,3,2)\\0&{\mbox{ otherwise }}\end{matrix}}\right.$

With this definition it turns out that

$\mathbf {u} \times \mathbf {v} =\epsilon _{ijk}{\hat {\mathbf {e} }}_{i}u_{j}v_{k}$

and

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

This will let us write many formulae more compactly.