Given $(\mathbb{R}^n, g)$, where $g$ is given by $g_{ij} = \dfrac{1}{x_i}\delta_{ij}$. Let $G = \begin{bmatrix}\dfrac{1}{x_i}\delta_{ij}\end{bmatrix}_{ij}$.
I need to find the Hessian of $f(x) = \sum\limits_{i = 1}^n x_i\ln(x_i), x \in \mathbb{R}^n$ with respect to this metric. I'm only one step away but I do not know how to continue because I don't have the right definition.
I know that the gradient with respect to metric $g$ is given by $\text{grad}(f) = G^{-1}\nabla f$, where $\nabla$ is the Eulidean gradient
Then: $\text{grad}(f) = [\text{diag}(1/x)]^{-1} \begin{bmatrix} \ln(x_1) +1 \\ \vdots \\ \ln(x_n)+1 \end{bmatrix} =\text{diag}(x) \begin{bmatrix} \ln(x_1) +1 \\ \vdots \\ \ln(x_n)+1 \end{bmatrix}$
Now all I have to do is to compute the Hessian. But I'm stuck and I have been searching for definition of Hessian with no avail.
On Wikipedia it says: https://en.wikipedia.org/wiki/Hessian_matrix#Generalizations_to_Riemannian_manifolds
${\displaystyle {\mbox{Hess}}(f)(X,Y)=\langle \nabla_X \text{grad}(f),Y \rangle} $ and ${\displaystyle {\mbox{Hess}}(f)(X,Y)=X(Yf)-df(\nabla _{X}Y)} $
But it offered no definition of $X,Y$ and $\nabla_X$. Similar search on MSE has no good result: How is the Hessian defined under a different metric + definition? If possible, I want to avoid calculating the Christoffel symbols.
How do I continue from here to calculate $\text{Hess}(f)$? Please assist! Any reference at a readable undergrad level would be greatly appreciated.
In the formula you quote, the expressions $X,Y$ are vector fields and $\nabla$ is a connection (in your case, this will be the Levi-Civita connection associated to the Riemannian metric). I don't think you'll be able to avoid computing the Christoffel symbols so the most straightforward approach will be to use the formula
$$ (\operatorname{Hess} f)(\partial_i, \partial_j) = \frac{\partial f}{\partial x^i \partial x^j} - \Gamma_{ij}^k \frac{\partial f}{\partial x^k}. $$
where the Christoffel symbols $\Gamma_{ij}^k$ of $g$ are given by the formula
$$ \Gamma_{ij}^k = \frac{1}{2} g^{im} \left( \frac{\partial g_{mi}}{\partial x^j} + \frac{\partial g_{mj}}{\partial x^i} - \frac{\partial g_{ij}}{\partial x^m}\right).$$
Here, $(g^{im})$ is the inverse matrix of $(g_{im})$ (so, in your case, $g^{im} = \delta^{im}x_i$).
Once you compute the Christoffel symbols, you only need to compute the first and second (Euclidean) partial derivatives of $f$ and then get the Hessian.