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, êi, 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.