can Maple invert partial derivatives

On Fri, 03 Nov 2006 16:45:36 +0000, Unruh

Well, this is OT, but I can't resist.

That is not quite true. It is not an if and only if situation. Given the three functions you suspect are partial derivatives, M, N, and P, say, then the condition that they really are the partial derivatives of some function f is called "exactness" of the differential 1-form Mdx + Ndy +Pdz.

The condition mentioned in this response is called "closedness" of that same 1-form, d-dy(M)=d-dx(N), etc. Exact forms are definitely closed, but closed forms are not always exact. If the domain where the functions are defined is what is called "simply-connected", then all closed 1-forms are exact, but the modern field of algebraic topology has grown up around the distinction between closedness and exactness in general.

The standard first example of this phenomenon is if M = -y-(x^2+y^2) and N = x-(x^2+y^2) (in the plane, or take P = 0). These are defined except where x=y=0, so except for the origin in the plane, or except for the z-axis in space. Mdx + Ndy is closed in that d-dy(M)=d-dx(N), but, as a function on the plane - origin (or space - z-axis), not exact.

As far as Maple finding the answer, certainly Maple can do the integrations indicated above. There may very well be a package smart enough to know the difference between closed and exact, in some cases, but I haven't seen it.

--

David L. Johnson

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