# Notation Conventions

This guide uses a single notation convention throughout all chapters:

1. Scalars use italic symbols, e.g. $p$, $\rho$, $T$, $Y_k$.
2. Vectors use arrow notation, e.g. $\vec{u}$, $\vec{A}$, $\vec{q}$.
3. Second-order tensors use double-underline notation, e.g. $\Matrix{\tau}$, $\Matrix{S}$.
4. Unit vectors use hats, e.g. $\hat{\vec{n}}$.
5. Dyadic products use $\otimes$, e.g. $\vec{u}\otimes\vec{u}$.

Face fluxes and area vectors use the stored `cl`-to-`cr` orientation. In a
balance for cell $P$, multiply each stored face contribution by its incidence
sign $\sigma_{Pf}$. A face-only subscript denotes the stored orientation;
a cell-face subscript, where used, denotes the quantity after this sign
conversion. The complete convention is defined in
{ref}`app_loci_face_orientation`.

For a transported algebraic row, ${}^{(\phi)}\!a_P$ is the diagonal in
cell $P$ and ${}^{(\phi)}\!a_{P,N}$ is its coupling to neighbor $N$.
The diagonal of the neighboring cell is ${}^{(\phi)}\!a_N$, a different
quantity. The complete known RHS is ${}^{(\phi)}\!b_P$; any pressure,
history, or source term removed from that RHS is identified explicitly.
For momentum, a vector RHS collects the three component RHS values.

Physical time levels use $n$ and $n+1$, while $k$ labels an outer nonlinear
iteration within a time step. A star identifies the provisional momentum
stage and a prime identifies a pressure-induced correction. A superscript
$\mathrm{pred}$ on a coefficient identifies the diagonal or RHS used by
the relaxed predictor; it is not another physical time level.


## Derivation Style Used In This Guide

For each major equation family (transport equations, pressure-correction equation, and time-discretization updates), we follow the same sequence:

1. Start from the continuous equation.
2. Integrate over the control volume (and time level, where needed).
3. State the approximation used (upwinding, linearization, neglected higher-order terms, etc.).
4. Show the algebraic form used in implementation.

When a derivation uses a closure or approximation, that assumption is stated explicitly next to the corresponding equation.