# 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 $$f$$, $$f(a)$$ denotes its value at $$a$$. $$Df$$ denotes its first derivative, and $$Df(a)$$ is the derivative evaluated at $$a$$, i.e

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

$$D^kf$$ denotes the $$k^{\text{th}}$$ derivative of $$f$$.

For a bi-variate function $$g(x,y)$$. $$D_1g$$ and $$D_2g$$ denote the partial derivatives of $$g$$ w.r.t the first and second variable respectively. In the classical notation this is equivalent to saying:

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

$$Dg$$ denotes the Jacobian of g, i.e.,

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

More generally for a multivariate function $$g:\mathbb{R}^n \longrightarrow \mathbb{R}^m$$, $$Dg$$ denotes the $$m\times n$$ Jacobian matrix. $$D_i g$$ is the partial derivative of $$g$$ w.r.t the $$i^{\text{th}}$$ coordinate and the $$i^{\text{th}}$$ column of $$Dg$$.

Finally, $$D^2_1g$$ and $$D_1D_2g$$ 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.