The tidal tensor is explicitly defined as the negative of the second partial derivatives of the potential: $T_{i j}=-\frac{\partial^2 \phi}{\partial x^2 \partial x^3}$. This is equivalent to stating that the tidal tensor is minus the Hessian matrix of the gravitational potential, since the Hessian matrix $H_{i j}$ is the matrix of second partial derivatives, $H_{i j}=\frac{\partial^2 \phi}{\partial x^2 \partial x^3}$. Furthermore, because the gravitational potential $\phi$ is a smooth function, the order of differentiation can be interchanged (by Clairaut's Theorem), which proves that the tidal tensor is symmetric (i.e., $T_{i j}=T_{j i}$).
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