What is the geometric meaning of inner product of $\nabla f$ and $\nabla f_x$?

Question

I am working on $3D$ imaging and I encoded one method through its Matlab code. The program used the following expression for a function $f:\Bbb{R}^3 \rightarrow \Bbb{R}$ modelizing the $3D$ image, $$ \nabla f.(\nabla f.\nabla f_x, \nabla f. \nabla f_y, \nabla f.\nabla f_z )$$ where $f_x$ denotes the $\dfrac{\partial f}{\partial x}$ and so on. What does that mean geometrically ? On the other hand, it is well known that the inner product of two vectors is connected with the angle between them. But I am confused about the above expression. So, I want to ask "Is there any book or article to describe the above expression? " or " What does that mean in terms of geometrical concepts ?".

Answer 1

I would write this expression as $$ \nabla^2 f(\nabla f, \nabla f) = \nabla f^T \ \nabla^2 f \ \nabla f.$$

Here $\nabla^2 f$ is the Hessian matrix of $f$; i.e. $(\nabla^2 f)_{ij} = \dfrac{\partial^2 f}{\partial x_i\partial x_j}$.

One way to interpret this is as the second derivative of $f$ in the direction $\nabla f$: we have

$$\nabla^2 f(\nabla f, \nabla f)|_\vec{x} = \frac{d^2}{dt^2}\Big|_{t=0} f(\vec x + t \nabla f).$$