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Partial derivative in terms of Kronecker delta and the Laplacian operator
No, the operator $\partial_i\partial_j$ cannot be expressed in terms of the Laplacian $\nabla^2$.
Indeed, take any real $a$ and $b$. If $\phi(x_1,\dots,x_n)=ax_1x_2+b(x_1^2+\dots+x_n^2)/(2n)$ for all …