Kronecker Delta

Relevant parts to questions...

• Substitution: ${\delta }_{ij}{a}_j=a_i$ - the delta “kills” itself and renames the dummy index.
• Collapse: ${\delta }_{ij}{\delta }_{jk}={\delta }_{ik}$ (chain of deltas merges).
• Trace: ${\delta }_{ii}=3$ (in 3D - the dimension of the space).
• Cartesian coordinate derivative: $\partial x_j/\partial x_i={\delta }_{ij}$.

The Kronecker delta, or substitution tensor, is defined by

$${\delta }_{ij}=\{\begin{array}{ll}1& \text{if }i=j\\ 0& \text{if }i\ne j\end{array}$$

where $i,j\in \{1,2,3\}$. Equivalent to the identity matrix:

$${\delta }_{ij}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$

Key Properties

The Kronecker delta acts like the substitution tensor::it replaces the repeated index with the free index, ${\delta }_{ij}{a}_j=a_i$, or symmetrically ${\delta }_{ji}{a}_i=a_j$. This works for any indexed object.

• Symmetric: ${\delta }_{ij}={\delta }_{ji}$.
• Tensor character: ${\delta }_{ij}$ is a rank-2 isotropic tensor - the same components in every Cartesian frame (${\delta }_{ij}^{\prime }=L_{ik}{L}_{j\ell }{\delta }_{k\ell }=L_{ik}{L}_{jk}={\delta }_{ij}$).
• Dot product: $\mathbf{a}\cdot \mathbf{b}=a_i{b}_i={\delta }_{ij}{a}_i{b}_j$.

Relates to the Alternating Tensor through

$${ϵ}_{ijk}{ϵ}_{i\ell m}={\delta }_{j\ell }{\delta }_{km}-{\delta }_{jm}{\delta }_{k\ell }$$

Collapse chains

${\delta }_{ij}{\delta }_{jk}{\delta }_{k\ell }={\delta }_{i\ell }$ - each delta “passes through” the chain renaming indices.

${\delta }_{ij}{\delta }_{ij}$: substitute $i\to j$ to get ${\delta }_{jj}=3$.

${\delta }_{ij}{a}_i{b}_j=a_i{b}_i=\mathbf{a}\cdot \mathbf{b}$.