General Mechanics/Index Notation

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Summation convention[edit]

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[edit]

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[edit]

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.