RE: Subvalues and Parameters in Differentiation and Usage Messages
- To: mathgroup at smc.vnet.net
- Subject: [mg67787] RE: Subvalues and Parameters in Differentiation and Usage Messages
- From: "David Park" <djmp at earthlink.net>
- Date: Fri, 7 Jul 2006 07:13:34 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Hi James,
It is very useful to be able to place parameters as subvalues. One often
doesn't want to differentiate with respect to the parameters. A person who
uses A LOT of that is Alfred Gray in his 'Modern Differential Geometry of
Curves and Surfaces with Mathematica'. For example he parametrizes a
circular hyperboloid of two sheets as...
hy2sheet[a_, c_][u_, v_] := {a Cosh[u]Cosh[v], a Sinh[u]Cosh[v], c Sinh[v]}
so if I used command completion on this (and I find command completion
extremely useful; it is what complements and makes practical the use of long
descriptive names that also seems to be a Mathematica style) I want to see
the complete command and not have the [u,v] part chopped off.
Furthermore everything works very well if we, for example, use an undefined
function f, differentiate and then afterwards substitute the actual function
as in...
D[f[a, c][u, v], u]
% /. f -> hy2sheet
giving
Derivative[1, 0][f[a, c]][u, v]
{a Cosh[v] Sinh[u], a Cosh[u] Cosh[v], 0}
So having subvalues in a head of a functions is perfectly acceptable in
Mathematica.
I am still looking for an answer as to the results of the following
statement.
D[foo[x][x], x]
foo[x]'[x] + foo'[x][x]
What is the mathematical justification of the second term? I don't think
that differentiation should act on the heads of expressions and am still
looking for a reason why it should. I don't see how the above will ever be a
useful answer and, in turn, it forecloses many useful constructions. But
there may be something I am missing.
In many cases the subvalue x is only an identifier. For example, we might
have a function, undefined at the moment, with x and y components,
f[x][x,y] and f[y][x,y]. It seems to me that it makes sense to have
differentiation work on the values in the last square brackets, but not to
touch subvalues in the head. I might want to differentiate these, say as
part of didactic exposition, and only later substitute the actual functions
for
f[x] and f[y].
So I'm not content with 'don't use subvalues' and am still looking for an
explanation. Either the Mathematica result is useful in an important class
of cases or the behavior should be changed.
David Park
djmp at earthlink.net
http://home.earthlink.net/~djmp/
From: James Gilmore [mailto:james.gilmore at yale.edu]
To: mathgroup at smc.vnet.net
Hi David,
With regard to your command completion comment, at least from my
perspective, separating parameters from variables, although useful in
some instances, does not seem to be in the spirit of the standardized
Mathematica functional language of the kernel. Anyone please correct me if
I'm wrong, but I don't know of a Kernel function with that syntax
f[n][x], they are all: f[n,x]. So I think we should at least try to
follow the WRI lead and maintain the standardized notation for our function.
As for the derivatives, writing the arguments in the standardized
notation removes any ambiguity.
In[39]:=
D[foo[p, x], x]
Out[39]=
Derivative[0, 1][foo][p, x]
In[40]:=
D[foo[x, x], x]
Out[40]=
Derivative[0, 1][foo][x, x] + Derivative[1, 0][foo][x, x]
Also, what exactly does the subscript x represent in D[Subscript[f,
x][x], x]? Is it the x component of a function? If so a "x" string
removes any ambiguity yet still looks the same.
In[45]:=
D[Subscript[f, "x"][x], x]
Out[45]=
Derivative[1][Subscript[f, "x"]][x]
Cheers
James Gilmore
- Follow-Ups:
- Re: RE: Subvalues and Parameters in Differentiation and Usage Messages
- From: "Carl K. Woll" <carlw@wolfram.com>
- Re: RE: Subvalues and Parameters in Differentiation and Usage Messages
- From: "Carl K. Woll" <carlw@wolfram.com>
- Re: RE: Subvalues and Parameters in Differentiation and Usage Messages