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 = (\begin{matrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{matrix}) 𝐁 = (\begin{matrix} B_{11} & B_{12} & B_{13} \\ B_{21} & B_{22} & B_{23} \\ B_{31} & B_{32} & B_{33} \end{matrix})

we can write similar expressions for matrix multiplications

(𝐀 𝐮)_{i} = \sum_{j = 1}^{3} A_{i j} u_{j} (𝐀 𝐁)_{i k} = \sum_{j = 1}^{3} A_{i j} B_{j k}

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} (𝐀 𝐮)_{i} = A_{i j} u_{j} (𝐀 𝐁)_{i k} = A_{i j} B_{j k}

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

𝐮 = u_{i} {\hat{𝐞}}_{i}

which can be convenient in a rotating coordinate system.

The Kronecker delta is

δ_{i j} = {\begin{matrix} 1 & i = j \\ 0 & i \neq j \end{matrix}

This is the standard way of writing the identity matrix.

Another useful quantity can be defined by

ϵ_{i j k} = {\begin{matrix} 1 & (i, j, k) = (1, 2, 3) or (2, 3, 1) or (3, 1, 2) \\ - 1 & (i, j, k) = (2, 1, 3) or (3, 2, 1) or (1, 3, 2) \\ 0 & otherwise \end{matrix}

With this definition it turns out that

𝐮 \times 𝐯 = ϵ_{i j k} {\hat{𝐞}}_{i} u_{j} v_{k}

and

ϵ_{i j k} ϵ_{i p q} = δ_{j p} δ_{k q} - δ_{j q} δ_{k p}

This will let us write many formulae more compactly.