TITLE: Third order term in Taylor Series
QUESTION [18 upvotes]: What is the third order term in the Taylor Series Expansion? I know it will just be third order partial derivatives but I want to know how is it expressed in a compact Matrix notation. For instance Jacobian for first order, Hessian for second order partial derivatives.
In other words, what is the third order term in the equation below? Thanks for your help!

REPLY [7 votes]: $\renewcommand{\Re}{\mathbb{R}}$Often Taylor expansions of functions $f:\Re^n\to\Re$ at a point $x$ are meant along a given direction $d\in\Re^n$. This facilitates a lot out understanding even for first-order expansions.
Let $f\in\mathcal{C}^3$ and define a function $\phi:\Re\to\Re$ given by $\phi(\tau) = f(x+\tau d)$; this is like a slice of $f$: it describes $f$ along the line $x+\tau d,\tau\in\Re$. Then $\phi$ is three times continuously differentiable and the third order expansion of $\phi$ about $\tau=0$ is
$$\begin{aligned}
\phi(\tau) = \phi(0) + \tau \phi'(0) + \tfrac{t^2}{2!}\phi''(0) + \tfrac{t^3}{3!}\phi'''(0) + o(t^3).
\end{aligned}$$
But $\phi'(0)$ is related to the directional derivative of $f$ at $x$ along the direction $d$ which is
$$\begin{aligned}
\phi'(0) &= \lim_{h\to 0}\frac{\phi(h) - \phi(0)}{h}\\
         &= \lim_{h\to 0}\frac{f(x+h d) - f(x)}{h}\\
         &= \langle \nabla f(x), d \rangle
\end{aligned}$$
Let us denote this by $\nabla_{d}f(x)$.
Similary, $\phi''(0)$ can be interpreted as the directional Hessian of $f$ at $x$ along the directions $d$ and $d$, that is
$$\begin{aligned}
\phi''(0) = \langle \nabla^2 f(x)d, d \rangle
\end{aligned}$$
Let us denote this by $\nabla^2_{d,d}f(x)$.
The term $\phi'''(0)$ - the "Tressian" as James S. Cook jokingly put it - is more difficult to represent. Indeed, it will be a tensor as Oren explained. However, we are merely interested in the "directional Tressian" of $f$ at $x$ along directions $d$, $d$ and $d$. This construct is actually used in the context of convex optimization theory and in particular the theory of self-concordant functions and is denoted by $\nabla^3_{d,d,d}f(x)$ and we may write
$$\begin{aligned}
\nabla^3_{d,d,d}f(x) = \langle \nabla^3 f(x)[d]d, d \rangle
\end{aligned}$$
where $\nabla^3 f(x)$ is the third-order gradient of $f$ at $x$ which, in my opinion, is best understood via its directional variant:
$$\begin{aligned}
\nabla^3f(x)[d] = \lim_{h\to 0} \frac{\nabla^2 f(x+\alpha d) - \nabla^2 f(x)}{h}
\end{aligned}$$
Here $\nabla^3f(x)[d]$ is a matrix - it is a directional Hessian. Essentially, $\nabla^3f(x)[d]$ describes how the Hessian of $f$ changes at $x$ along the direction $d$.