=============== 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.