ceres-solver / ceres-solver / 12263e28305a2a43b6c6a6b4f7f76814ab7ffa5f / . / docs / source / spivak_notation.rst

.. default-domain:: cpp | |

.. cpp:namespace:: ceres | |

.. _chapter-spivak_notation: | |

=============== | |

Spivak Notation | |

=============== | |

To preserve our collective sanities, we will use Spivak's notation for | |

derivatives. It is a functional notation that makes reading and | |

reasoning about expressions involving derivatives simple. | |

For a univariate function :math:`f`, :math:`f(a)` denotes its value at | |

:math:`a`. :math:`Df` denotes its first derivative, and | |

:math:`Df(a)` is the derivative evaluated at :math:`a`, i.e | |

.. math:: | |

Df(a) = \left . \frac{d}{dx} f(x) \right |_{x = a} | |

:math:`D^kf` denotes the :math:`k^{\text{th}}` derivative of :math:`f`. | |

For a bi-variate function :math:`g(x,y)`. :math:`D_1g` and | |

:math:`D_2g` denote the partial derivatives of :math:`g` w.r.t the | |

first and second variable respectively. In the classical notation this | |

is equivalent to saying: | |

.. math:: | |

D_1 g = \frac{\partial}{\partial x}g(x,y) \text{ and } D_2 g = \frac{\partial}{\partial y}g(x,y). | |

:math:`Dg` denotes the Jacobian of `g`, i.e., | |

.. math:: | |

Dg = \begin{bmatrix} D_1g & D_2g \end{bmatrix} | |

More generally for a multivariate function :math:`g:\mathbb{R}^n | |

\longrightarrow \mathbb{R}^m`, :math:`Dg` denotes the :math:`m\times | |

n` Jacobian matrix. :math:`D_i g` is the partial derivative of | |

:math:`g` w.r.t the :math:`i^{\text{th}}` coordinate and the | |

:math:`i^{\text{th}}` column of :math:`Dg`. | |

Finally, :math:`D^2_1g` and :math:`D_1D_2g` have the obvious meaning | |

as higher order partial derivatives. | |

For more see Michael Spivak's book `Calculus on Manifolds | |

<https://www.amazon.com/Calculus-Manifolds-Approach-Classical-Theorems/dp/0805390219>`_ | |

or a brief discussion of the `merits of this notation | |

<http://www.vendian.org/mncharity/dir3/dxdoc/>`_ by | |

Mitchell N. Charity. |