********************
Notation and Symbols
********************

.. _common_notation:
.. list-table:: Common Notation
   :header-rows: 1
   :widths: 10 25
   :align: center

   * - Notation
     - Meaning
   * - :math:`x`
     - a scalar
   * - :math:`\vec{x}`
     - a vector
   * - :math:`x^y`
     - a scalar raised to a power, where :math:`y` is a scalar
   * - :math:`\vec{x}^c`
     - a vector in the coordinate system :math:`c`
   * - :math:`\vec{x}_{B/A}`
     - a vector from point A to point B ("B with respect to A")
   * - :math:`{^r \dot{\vec{x}}}`
     - the derivative of a vector taken in reference frame
       :math:`\mathcal{F}_r`
   * - :math:`x_k`
     - a variable at index :math:`k` of a sequence of length :math:`K`
   * - :math:`x^{(n)}`
     - element :math:`n` of a set of :math:`N` elements
   * - :math:`\mat{X}_{M \times N}`
     - a matrix with :math:`M` rows and :math:`N` columns
   * - :math:`\mat{X}^z`
     - a matrix exponential, where :math:`z` is a scalar
   * - :math:`\left| x \right|`
     - absolute value of a scalar
   * - :math:`\norm{\vec{x}}`
     - Euclidean norm of a vector
   * - :math:`\left| \mat{X} \right|`
     - determinant of a matrix
   * - :math:`\mat{C}_{b/a}`
     - the directed cosine matrix that transforms vectors from coordinate
       system :math:`a` into coordinate system :math:`b`
   * - :math:`\vec{q}_{b/a}`
     - a quaternion that encodes the relative orientation of coordinate system
       :math:`b` relative to coordinate system :math:`a`
   * - :math:`\vec{\omega}_{b/a}`
     - angular velocity vector of frame :math:`\mathcal{F}_b` with respect to frame
       :math:`\mathcal{F}_a`
   * - :math:`f(\cdot)`, :math:`func(\cdot)`, etc
     - functions, where ``f``, ``func``, can be any identifier

Another notation which is useful when building systems of equations involving
matrices is the *cross-product matrix operator*, so that
:math:`\crossmat{\vec{v}} \vec{x} \equiv \vec{v} \times \vec{x}`:

.. _crossmat:
.. math::

   \crossmat{\vec{v}} \defas
      \begin{bmatrix}
         0 & -v_3 & v_2\\
         v_3 & 0 & -v1\\
         -v_2 & v_1 & 0
      \end{bmatrix}

By their nature, vectors require the most intricate notation, since a fully
specified vector might include all of:

1. A reference frame

2. A coordinate system

3. A fixed point (if it's a bound vector)

For simplicity, :numref:`common_notation` only shows examples of each distinct
element of a vector encoding. In practice, vectors may appear quite complex;
for some realistic examples taken from
:cite:`stevens2015AircraftControlSimulation`:

.. math::

   \begin{aligned}
   \vec{p}_{A/B} &\defas
      \text{the position of the point A with respect to point } B \\
   \vec{v}_{A/i} &\defas
      \text{the velocity vector of a point } A \text{ in frame } \mathcal{F}_i \\
   ^b \dot{\vec{v}}_{A/i} &\defas
      \text{the vector derivative of } \vec{v}_{A/i} \text{ taken in frame } \mathcal{F}_b \\
   \vec{v}^c_{A/i} &\defas
      \text{array of components of } \vec{v}_{A/i} \text{ in coordinate system } c \\
   ^b \dot{\vec{v}}^c_{A/i} &\defas
      \text{components in coordinate system } c \text{ of the derivative taken in frame } \mathcal{F}_b
   \end{aligned}