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mixed second derivatives

Here's a little tidbit that I'm sure is obvious to many wizards. But
it almost got me in hot water in a problem I was working on
yesterday: When can you *not* reverse the order of differentiation of
two derivatives, ie. when is d^2f/dxdy not equal to d^2f/dydx?

The rough answer is: When x and y are not independent variables. A
more rigorous answer, which I'll leave as a simple exercise for the
reader to verify, follows from:

d/dx(df/dy) - d/dy(df/dx) = d/dx(dx/dy) * df/dx

and hence you cannot interchange the order of differentiation if y is
a nonlinear function of x (ignoring the trivial possibility that f is
a constant). Carl
Carl E. Mungan, Asst. Prof. of Physics 410-293-6680 (O) -3729 (F)
U.S. Naval Academy, Stop 9C, Annapolis, MD 21402-5040