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.