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54 lines
1.7 KiB
54 lines
1.7 KiB
3 years ago
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.. default-domain:: cpp
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.. cpp:namespace:: ceres
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.. _chapter-spivak_notation:
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===============
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Spivak Notation
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===============
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To preserve our collective sanities, we will use Spivak's notation for
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derivatives. It is a functional notation that makes reading and
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reasoning about expressions involving derivatives simple.
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For a univariate function :math:`f`, :math:`f(a)` denotes its value at
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:math:`a`. :math:`Df` denotes its first derivative, and
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:math:`Df(a)` is the derivative evaluated at :math:`a`, i.e
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.. math::
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Df(a) = \left . \frac{d}{dx} f(x) \right |_{x = a}
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:math:`D^kf` denotes the :math:`k^{\text{th}}` derivative of :math:`f`.
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For a bi-variate function :math:`g(x,y)`. :math:`D_1g` and
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:math:`D_2g` denote the partial derivatives of :math:`g` w.r.t the
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first and second variable respectively. In the classical notation this
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is equivalent to saying:
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.. math::
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D_1 g = \frac{\partial}{\partial x}g(x,y) \text{ and } D_2 g = \frac{\partial}{\partial y}g(x,y).
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:math:`Dg` denotes the Jacobian of `g`, i.e.,
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.. math::
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Dg = \begin{bmatrix} D_1g & D_2g \end{bmatrix}
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More generally for a multivariate function :math:`g:\mathbb{R}^n
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\longrightarrow \mathbb{R}^m`, :math:`Dg` denotes the :math:`m\times
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n` Jacobian matrix. :math:`D_i g` is the partial derivative of
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:math:`g` w.r.t the :math:`i^{\text{th}}` coordinate and the
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:math:`i^{\text{th}}` column of :math:`Dg`.
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Finally, :math:`D^2_1g` and :math:`D_1D_2g` have the obvious meaning
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as higher order partial derivatives.
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For more see Michael Spivak's book `Calculus on Manifolds
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<https://www.amazon.com/Calculus-Manifolds-Approach-Classical-Theorems/dp/0805390219>`_
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or a brief discussion of the `merits of this notation
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<http://www.vendian.org/mncharity/dir3/dxdoc/>`_ by
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Mitchell N. Charity.
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