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

$\delta_{ij}= \left\{ \begin{matrix} 1 & i=j\\ 0 & i\ne j \end{matrix} \right.$

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.