What is the einstein notation for the hessian matrix

Question

I am taking a course on general relativity and our teacher asked us to explore writing multivariable calculus objects in einstein/tensor form, is this the right one for the hessian matrix: $ H_{f}= fD^{2}_{x^{i}x^{j}}\vec{e_{i}}\vec{e_{j}} $?

Answer 1

The function has to be written behind the differential operator. One would rather write

$(H_f)_{i,j} = D^2_{x_i,x_j} f$, or to write as a tensor, $H_f = (D^2_{x_i,x_j} f) e_i\otimes e_j$.