Two points about this slide:
1) I am not sure that the Hessian is indeed always symmetric. Intuitively, it makes sense that the partials commute for smooth functions, but I suspect they don't necessarily do so for all functions.
2) I think the second column of partials may be missing their "f"'s.
For the Hessian Matrix to be symmetric we would need the second partial derivatives to be continuous on the domain. Under this circumstance, we can use the symmetry of second derivatives to show that Hessian is indeed symmetric. I guess for most geometric and physics functions we can assume them to be smooth and sufficiently differentiable to assume that the Hessian is always symmetric.
1) As @sbhilare suggests: as long as the function is $C^2$ (i.e., it has continuous 2nd derivatives) then equality of mixed partials holds, by Clairaut's theorem. (If not, you have to talk about what you mean by the second partial derivative!)
2) Thanks; now fixed for next year. :-)