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 .. 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 _ or a brief discussion of the merits of this notation _ by Mitchell N. Charity.