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MathGroup Archive 2006

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Re: Total Derivative and Output Display Question

  • To: mathgroup at smc.vnet.net
  • Subject: [mg65142] Re: Total Derivative and Output Display Question
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Wed, 15 Mar 2006 06:30:31 -0500 (EST)
  • Organization: The University of Western Australia
  • Sender: owner-wri-mathgroup at wolfram.com

Matt wrote:

> I have been experimenting with the Dt[] function in Mathematica.
> The reason for this is that as I review or work with new topics in
> math, I try to see how the same results achieved by hand can or will be
> realized in Mathematica.  I recently needed to review the concept of
> the total derivative, and as is my usual practice, to ensure I
> understood what I was doing, I came up with an example that would be a
> challenge, that is, a challenge for me.
>   Here's what I did on paper (the actual example was two levels deeper
> than this, but this should suffice for the sake of argument):
> 
> z = f(x,y);
> x = g(r,s);
> y = h(r,s);
> r = a(t,u);
> s = b(t,u);
> 
> What is the total (partial) derivative of z with respect to 't'?
> 
> On paper, I worked this out to (all derivatives should be considered as
> partial):
> 
> dz/dt = dz/dx(dx/dr dr/dt + dx/ds ds/dt) + dz/dy(dy/dr dr/dt + dy/ds
> ds/dt).

Here is another approach:

  D[z /. z -> z[x, y] /. {x -> x[r, s], y -> y[r, s]} /. 
    {r -> r[t], s -> s[t]}, t]

  % /. {Derivative[1, 0][f_][__] :> HoldForm[D[f, r]], 
        Derivative[0, 1][f_][__] :> HoldForm[D[f, s]],
        Derivative[1][f_][_] :> HoldForm[D[f, t]]}

> As you can see, there are no f's, g's, h's, a's, or b's, because we all
> understand that when we state something like z = f(x, y), we are
> stating that z is the dependent variable and x and y are the
> independent variables and 'f' is just part of the notation indicating
> that z is a function of x and y.

What you are after is an _implicit_ method of declaring the variable 
dependency of a function; no such functionality is 
built-in.

> As those of you who download it will see, there are numerous problems
> with my approach.  The first is that it appears overly burdensome to
> convert from Mathematica's default output form, to the more
> 'traditional' mathematical form.  Secondly, even after I convert, I end
> up with the differential operators being in the wrong order, and I'm
> not really sure about how to avoid that.

There are no differential operators in your final expression, only 
derivatives of functions -- so the order of terms does not matter here. 
Also, functionally the original output is perfectly suitable for further 
calculations, so the "preferred" simpler typeset form is really for 
aesthetics. 

Enter x-1 into Mathematica. StandardForm re-orders this to -1 + x. If 
you prefer x-1 convert to TraditionalForm.

Perhaps the discussion in "Structure and Interpretation of Classical 
Mechanics" by Gerald Jay Sussman and Jack Wisdom, available online at 
http://mitpress.mit.edu/SICM/ is of interest. In particular, have a read 
of the Preface at

  http://mitpress.mit.edu/SICM/book-Z-H-5.html

Cheers,
Paul

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul


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