Suffix Notation

Relevant parts to questions...

• Converting everything into suffix form by applying free and dummy indices.
• Using $\mathbf{a}\cdot \mathbf{b}=a_i{b}_i$, both directions.
• Using $(\mathbf{AB}{)}_{ij}=A_{ik}{B}_{kj}$, $(A^T{)}_{ij}=A_{ji}$, and $\mathrm{Tr}(\mathbf{A})=A_{ii}$.
• Einstein convention: any index that appears exactly twice in a term is summed over $1,2,3$.

Suffix Notation (or Index Notation) drops the explicit $\Sigma$ symbol via the Einstein summation convention - repeated indices are implicitly summed. Two index types appear:

• Dummy indices: appear exactly twice in a term, summed over $1,2,3$ - arbitrary name, no presence in result, e.g., ${\sum }_{j=1}^3{a}_j{b}_j\to \boxed{a_j{b}_j}$.
• Free indices: appear exactly once in a term, label components of the result - same name on every term in an equation, e.g., the $i$-th component of a vector.

Index-counting sanity checks (always verify):

1. Every term must have the same free indices.
2. No index appears more than twice in a single term.
3. Number of free indices = rank of the expression.

Suffix Notation Dictionary

Vector/matrix form	Suffix form	Notes
Dot product $\mathbf{a}\cdot \mathbf{b}$	$a_i{b}_i$	Scalar
Cross product $(\mathbf{a}\times \mathbf{b}{)}_i$	${ϵ}_{ijk}{a}_j{b}_k$	Vector
Matrix product $(\mathbf{AB}{)}_{ij}$	$A_{ik}{B}_{kj}$	Rank 2
Transpose $({\mathbf{A}}^T{)}_{ij}$	$A_{ji}$	Swap indices
Trace $\mathrm{Tr}(\mathbf{A})$	$A_{ii}$	Scalar
Coordinate derivative $\partial x_j/\partial x_i$	${\delta }_{ij}$	Cartesian only

Converting to Suffix Form

1. Introduce free index for each free slot of the result (e.g., $i$ for vector identities).
2. Distribute the free index across each vector/tensor in the expression.
3. Introduce a fresh dummy index for each scalar sub-expression (dot product, contraction).

Example

Convert $\mathbf{u}+(\mathbf{a}\cdot \mathbf{b})\mathbf{v}=(\mathbf{a}\cdot \mathbf{a})(\mathbf{b}\cdot \mathbf{v})\mathbf{a}$:

1. Apply free index $i$: $u_i+(\mathbf{a}\cdot \mathbf{b})v_i=(\mathbf{a}\cdot \mathbf{a})(\mathbf{b}\cdot \mathbf{v})a_i$;
2. Introduce a fresh dummy index for each scalar dot product: $\boxed{u_i+a_j{b}_j{v}_i=a_k{a}_k{b}_{\ell }{v}_{\ell }{a}_i}$.

Note that we use different dummy indices for each scalar - never reuse a dummy index within the same term.