Notation of Hessian? I looked here, and couldn't find an answer:
http://en.wikipedia.org/wiki/Hessian_matrix
I am looking for a way to denote the Hessian at a certain value.
I have a function $f(x)$ where $x \in \mathbb{R}^d$, and I want to somehow denote
the second derivative of $f$ with respect to $x_i$ and $x_j$ in point $x^{\ast}$.
The second derivative would be just $$\frac{\partial^2f}{\partial x_i \partial x_j}$$, but how would you denote the evaluation at point $x^{\ast}$, especially using latex?
 A: You seem to be asking at two different questions, which are reflected in the current answers.
If you want to denote an entry of the Hessian matrix (this is a second partial derivative of a function) at a particular point $p$, there are many ways to do this:
$$
\frac{\partial^2 f}{ \partial x_i \partial x_j}(p)
$$
or $$\left.\frac{\partial^2 f}{ \partial x_i \partial x_j}\right|_p$$
or $$f_{i,j} (p)$$ 
might all be appropriate depending on context.
If you want a way to talk about the Hessian matrix at a particular point, then using $$H(f)(p)$$ or $$H_f(p)$$ might be appropriate
If you view the Hessian as a quadratic form, then as Ivo Terek mentions, you might want apply the Hessian to a vector at a point and write
$$H_f(p,v)$$ or something similar.
In my opinion the "best" perspective on the Hessian is as a bilinear form which approximates the change in the directional derivative of $f$ in direction $v_1$ which results from moving in direction $v_2$.  That is, I would write
$$H_f(p)(v_1,v_2) \approx Df_{p+v_2}(v_1)-Df_p(v_1)$$.
From this perspective, one can recover the second partials from the bilinear map as
$$\left.\frac{\partial^2 f}{\partial x_i \partial x_j}\right|_p = H_f(p)(e_i,e_j)$$
where $e_k$ are the standard basis vectors.
The bottom line is:  decide on a notation, explain it, and use it consistently
A: I also saw, which I particularly like, $\mathrm{Hess} \ f_p$ for the quadratic form. This way, we would have: $$\mathrm{Hess} \ f_p({\bf v}) = \begin{pmatrix} v_1 & \cdots & v_n\end{pmatrix}\begin{pmatrix}\frac{\partial^2f}{\partial x_1 \partial x_1} & \cdots & \frac{\partial^2f}{\partial x_1 \partial x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial^2f}{\partial x_n \partial x_1} & \cdots & \frac{\partial^2f}{\partial x_n \partial x_n}\end{pmatrix} \begin{pmatrix} v_1 \\ \vdots \\ v_n\end{pmatrix}.$$
For the determinant, $|\mathrm{Hess} \ f_p|$. 
A: Keep it simple $$D^2f(x)$$ that's the standard in Evans since the Hessian is the gradient of the gradient of $f$.
A: I would write $H(f)(x^{*})$, where $H$ is the Hessian, or, if you really like the "partials" notation, I'd write
$$
\frac{\partial^2 f}{\partial x_i \partial x_j}(x^{*})
$$
or
$$
\frac{\partial^2 f}{\partial x_i \partial x_j}(x_1^{*}, \ldots, x_n^{*}).
$$
More honestly, I'd look at Milnor's Morse Theory and copy whatever he did. 
A: It is up to you to define your own notation however you choose. 
Personally, I use $J_f$ and $H_f$ to denote the Jacobian and Hessian matrices of $f$ respectively.
I then use $|J_f|$ and $|H_f|$ for their determinants. Hence $|H_f|(p)$ would be the determinant of the Hessian matrix of $f$ evaluated at the point $x=p$. 
If you want to use the "evaluated at notation", e.g.
$$\left. \frac{\partial f}{\partial x} \right|_{x=p}$$
then you can use \left. \right|_{x=p} to get the line. The \left( \right) type commands give enlarged brackets that match the height of the text. Using \left. or \right. leaves out a bracket you don't want. They still need to come in parts though, i.e. \left\{ \right. and \left. \right) are both fine, but don't try \left\{ or \right) on their own.
A: My professor teachs me that the hessian of a funcion $f$ is written as $\Delta H_f(x,y)$.
