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?
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
Keep it simple $$D^2f(x)$$ that's the standard in Evans since the Hessian is the gradient of the gradient of $f$.
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, $|\mathrm{Hess} \ f_p|$.
My professor teachs me that the hessian of a funcion $f$ is written as $\Delta H_f(x,y)$.
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.
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.
