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# Re: mixed second derivatives

• From: Jack Uretsky <jlu@HEP.ANL.GOV>
• Date: Tue, 31 Aug 2004 18:31:44 -0500

if y = y(x), then I get:

[(d/dy),(d/dx)] = (y''/(y')^2)(d/dx) Did I slip up somewhere?

Now why do these look like elements of a Lie algebra? Does
this have anything to do with topics treated in Ince <Differential
Equations>?

Regards,
Jack

On Tue, 31 Aug 2004, Carl E. Mungan wrote:

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

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
mailto:mungan@usna.edu http://usna.edu/Users/physics/mungan/

--
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