When is a matrix a hessian?

quasar987

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Apparently it is a well-known fact that if [itex]G(x)=(G_{ij}(x_1,\ldots,x_n))[/itex] is a smooth nxn matrix-valued function such that [itex]G_{ij,k}=G_{ik,j}[/itex] for all i,j,k, then there exists a smooth function g such that Hess(g)=G; i.e. [itex]g_{,ij}=G_{ij}[/itex]. ([itex]f_{,k}[/itex] denotes partial differentiation with respect to the kth variable.)

I believe I can construct the solution explicitly in the n=2 case, but I'm not sure how to generalize my argument. Is there an argument to be made about the existence of a solution to this overdetermined system of PDE? Thx!
 

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