Given a smooth function $u$, what is the meaning of $\nabla u \otimes D^2 u$ (where $D^2$ is the Hessian matrix)?

Question

Given a smooth function $u$, how does one compute $\nabla u \otimes D^2 u$ (where $D^2$ is the Hessian matrix and $\otimes$ is the tensor product) and what is its meaning?

Answer 1

I'm going to use index notation (and in the process assume Cartesian coordinates) to elucidate.

In index form, those objects look like:

$$\nabla u = \frac{\partial u}{\partial x_i}, \quad D^2 u = \frac{\partial^2 u}{\partial x_j \partial x_k}$$

Taking the tensor product of both of them just means tacking everything together:

$$\nabla u \otimes D^2 u = \frac{\partial u}{\partial x_i}\frac{\partial^2 u}{\partial x_j \partial x_k}$$

This is a third-order tensor as noted by the three different indices. If you're trying to get some insight into what this object might represent, consider the following representation:

$$\nabla u \otimes D^2 u = \frac{\partial}{\partial x_i}\left(\frac{1}{2}\frac{\partial u}{\partial x_k}\frac{\partial u}{\partial x_j}\right)$$

In coordinate-free notation, this would appear to indicate:

$$\nabla u \otimes D^2 u = \frac{1}{2}\nabla\left(\nabla u\otimes \nabla u\right)$$

Hope this is helpful! Feel free to call me out if the third expression is wrong.