Hessian matrix of the Newtonian potential is zero?

So I'm looking at the hessian of the Newtonian potential:



[itex] \partial^2\phi / \partial x_i \partial x_j [/itex]



Using the fact that (assuming the mass is constant):



[itex] F = m \cdot d^2 x / d t^2 = - \nabla \phi [/itex]



This implies:



[itex] \partial^2\phi / \partial x_i \partial x_j = -m \cdot \frac{\partial}{\partial x_j} (d^2 x_i / d t^2) = -m \cdot \frac{\partial}{\partial x_j} (\partial^2 x_i / \partial t^2)[/itex]



As we can swap the total derivatives for partial derivatives since for Cartesian coordinates:



[itex] \partial x_i / \partial x_j = \delta_{ij} [/itex]



Using the fact that we can swap the order of differentiation for mixed partials (assuming continuity of the partial derivatives) we obtain:



[itex] \partial^2\phi / \partial x_i \partial x_j = -m \cdot \partial^3 x_i / \partial x_j \partial t^2 = -m \cdot \frac{\partial}{\partial t^2} \partial x_i / \partial x_j = -m \cdot 0 = 0 [/itex]



Hence I obtain the result that the hessian of the Newtonian potential is zero which can't possibly be correct but I can't find the error in my calculation.



Any help would be much appreciated :)