Bug in D (or Derivative) ( Was: Re: DSolve bug)

• To: mathgroup at smc.vnet.net
• Subject: [mg97919] Bug in D (or Derivative) ( Was: Re: DSolve bug)
• From: schochet123 at gmail.com
• Date: Wed, 25 Mar 2009 05:46:54 -0500 (EST)
• References: <612696379.19321237704569790.JavaMail.root@mmm.inp.demokritos.gr>

```The problem is not just that D[u[x][y],x,y] and D[u[y][x],y,x] yield
apparently different results but that the results Mathematica gives
for both of them are wrong.

For example, define
usub=u -> (Function[{x}, Function[{y}, E^(a x + b y)]]);

Substituting and then differentiating (in either order) gives correct
results (as several posters pointed out):
D[u[x][y]/.usub,x,y] yields   a b E^(a x + b y)
However D[u[x][y],x,y]/.usub yields the incorrect answer   a^2 E^(a x+
b y)
Taking the derivative in the opposite order and then substituting
yields an expression that does not even reduce to a specific function
of x and y.

The source of this bug seems to be the following bug in in D (or
Derivative): According to Mathematica, D[u',u] yields u''. However,
(Homework for Calculus 1: use the chain rule to prove this.)

The way Mathematica obtains its incorrect result is quite amusing, As
Trace shows, Mathematica evaluates D[u',u] by noting that the FullForm
of u' is Derivative[1][u], whose derivative with respect to u is (of
course ;) ) Derivative[1]'[u].
Mathematica then decides that Derivative[1]'  is Derivative[2],
yielding Derivative[2][u], i.e., u''.

Steve

On Mar 22, 12:50 pm, DrMajorBob <btre... at austin.rr.com> wrote:
> More to the point, using Bob's example:
>
> Clear[f]
> f[a_][x_] := Sin[a*x]
> D[f[x][y], x, y] - D[f[x][y], y, x]
>
> 0
>
> As you can see, both derivatives ARE the same. Here's a better test, I =

> think (since a and x were used symmetrically, above):
>
> Clear[f]
> f[a_][x_] := Sin[a]^x
> diff = D[f[x][y], x, y] - D[f[x][y], y, x]
>
> 0
>
> But let's try it Sotirios' way, keeping f symbolic for a bit:
>
> Clear[f]
> raw=D[f[x][y],x,y]-D[f[x][y],y,x];
> Block[{g},g[a_][x_]:=Sin[a]^x;raw/.f->g]//ExpandAll//Simplify
>
> -(Log[Sin[x]]^2 Sin[x]^#1& (g^\[Prime])[x])[y]+(g^\[Prime]\[Prime])[x][y]
>
> Now Mathematica can't equate the two derivatives.
>
> But that's a problem with D, not withDSolve.
>
> Note that Dt does better, too:
>
> Clear[f]
> raw = Dt[f[x][y], x, y] - Dt[f[x][y], y, x]
>
> 0
>
> DSolve'ssolution IS correct. It's D that may not be all it should be.
>
> Bobby
>
> On Sun, 22 Mar 2009 01:49:29 -0500, Sotirios Bonanos
>
>
>
> <sbon... at inp.demokritos.gr> wrote:
>
> > Your example gives two different ways of referring to the same function=

> > (Sin[a*x]). But I want to be able to use the arbitrary function in the =

> > solution given byDSolve(and its derivatives) in other expressions.
> > This I cannot do because the derivatives D[f1[x][y], x, y], D[f1[x][y],=

> > y, x] are not equal:
>
> > Clear[f1, f2]
>
> > {D[f1[x][y], x, y], D[f1[x][y], y, x]}
>
> > That's why I claim, if it is not abug, it is an unfortunate choice of =

> > representation!
>
> > Sotirios Bonanos
>
> > ----- "Bob Hanlon" wrote:
> >> It is just an alternate representation
>
> >> Clear[f1, f2]
>
> >> f1[a_][x_] := Sin[a*x]
>
> >> f2[a_, x_] := Sin[a*x]
>
> >> f1[c][t] == f2[c, t]
>
> >> True
>
> >> {Plot3D[f1[a][x], {x, 0, 2 Pi}, {a, 1, 3}],
> >> Plot3D[f2[a, x], {x, 0, 2 Pi}, {a, 1, 3}]}
>
> >> Some people prefer the f[a][x] representation to explicitly separate o=
ut
> >> parameter(s) from argument(s)
>
> >> If you prefer
>
> >>DSolve[D[F[x, y, z], x, y] == 0, F[x, y, z], {x, y, z}]
>
> >> {{F(x,y,z)->Subscript[c, 1][z][x]+Subscript[c, 2][z][y]}}
>
> >> % /. f_[arg1_][arg2_] :> f[arg2, arg1]
>
> >> {{F(x,y,z)->Subscript[c, 1][x,z]+Subscript[c, 2][y,z]}}
>
> >> Bob Hanlon
>
> >> On Sat, Mar 21, 2009 at 10:25 AM , Sotirios Bonanos wrote:
>
> >> > Hello,
> >> > I have encountered the followingbuginDSolve:
> >> >DSolve[D[F[x, y, z], x, y] == 0, F[x, y, z], {x, y, z}]
> >> > gives {{F[x, y, z] -> C[1][z][x] + C[2][z][y]}}
> >> > instead of {{F[x, y, z] -> C[1][x, z] + C[2][y, z]}}
> >> > I don't know if this has been fixed in Mathematica 7, but it is
> >> > present in versions 5 and 6.
> >> > S. Bonanoshttp://www.inp.demokritos.gr/~sbonano/
>
> --
> DrMajor... at bigfoot.com

```

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