Notation of partial derivative

Question

I want to write partial derivatives of functions with many arguments. Why is it that when I type

f[x,y] ctrl+6 (0,1)

it turns out to be bad syntax? The output of

D[f[x,y],y]

looks very much like f with a superscript (0,1).

Answer 1

For a start, f[x,y]^(0,1) isn't the same as f^(0,1)[x,y].

But the real reason is that these expressions are very different in meaning, as revealed by their FullForm:

D[f[x, y], y] // FullForm

Derivative[0,1][f][x,y]

versus (and I had to use a simple symbolic expression as the exponent to show what was going on:

f[x,y]^z//FullForm

Power[f[x,y],z]

Similarly, Derivative doesn't correspond to Superscript. They are syntactically different despite the visual similarities.

Stick with D[f[x,y],y] and so on. If you need the vector derivative, you can use the syntax:

 D[f,{{x1,x2,x3...}}]

as described in the documentation.

Answer 2

Verbeia is right. An alternative notation is to use escpdesc which gives a partial derivative; thus, typing escpdesc ctrl-t followed by f[x,t] will give the derivative of f with respect to its second argument.

For instance, this is a valid way to specify a differential equation:

This is closer to what you're after than D[f[x,t],t], for instance.

Answer 3

I have a function called "AbleitungsForm" (Ableitung is german for Derivative) which is based on an answer I found here in SE. I coudn't find the original answer. It looks like this:

AbleitungsForm::ON = "AbleitungsForm with Options \"AuchStandard\[Rule]`1`\" and \"MitArgumenten\[Rule]`2`\" is activ.";
AbleitungsForm::OFF = "AbleitungsForm has been deactivated.";
abFOpts={}

SyntaxInformation[AbleitungsForm]={"ArgumentsPattern"->{_,OptionsPattern[]}};
Options[AbleitungsForm]={AuchStandard->True,MitArgumenten->False};
AbleitungsForm[On,opt:OptionsPattern[]] :=
(Quiet[AbleitungsForm[Off]];
 abFOpts={OptionValue[AuchStandard],OptionValue[MitArgumenten]};
  If[OptionValue[AuchStandard] === True,
    If[OptionValue[MitArgumenten] === False,
     Derivative /: 
      MakeBoxes[Derivative[inds__][g_][vars__Symbol], 
       form : TraditionalForm | StandardForm | DAFX] :=
      Module[{bb, dd, sp},
        MakeBoxes[dd, _] ^= 
           If[Length[{inds}] == 1, "\[DifferentialD]", "\[PartialD]"];
        MakeBoxes[sp, _] ^= "\[ThinSpace]";
        bb /: MakeBoxes[bb[x__], _] := RowBox[Map[ToBoxes[#] &, {x}]];
        FractionBox[ToBoxes[bb[dd^Plus[inds], g]], 
          ToBoxes[Apply[bb, 
            Riffle[Map[bb[dd, #] &, Select[({vars}^{inds}), (# =!= 1 &)]],
              sp]]]]],
     Derivative /: 
      MakeBoxes[Derivative[inds__][g_][vars__Symbol], 
       form : TraditionalForm | StandardForm | DAFX] :=
      Module[{bb, dd, sp, vd},
        MakeBoxes[dd, _] ^= 
           If[Length[{inds}] == 1, "\[DifferentialD]", "\[PartialD]"];
        MakeBoxes[sp, _] ^= "\[ThinSpace]";
        vd[f_, v__, fmt_] := DisplayForm@ToBoxes[f[v], fmt];
        bb /: MakeBoxes[bb[x__], _] := RowBox[Map[ToBoxes[#] &, {x}]];
        FractionBox[ToBoxes[bb[dd^Plus[inds], vd[g, vars, form]]], 
          ToBoxes[Apply[bb, 
           Riffle[Map[bb[dd, #] &, Select[({vars}^{inds}), (# =!= 1 &)]],
             sp]]]]]
    ],
    If[OptionValue[MitArgumenten] === False,
     Derivative /: 
      MakeBoxes[Derivative[inds__][g_][vars__Symbol], 
       form : TraditionalForm | DAFX] :=
      Module[{bb, dd, sp},
        MakeBoxes[dd, _] ^= 
           If[Length[{inds}] == 1, "\[DifferentialD]", "\[PartialD]"];
        MakeBoxes[sp, _] ^= "\[ThinSpace]";
        bb /: MakeBoxes[bb[x__], _] := RowBox[Map[ToBoxes[#] &, {x}]];
        FractionBox[ToBoxes[bb[dd^Plus[inds], g]], 
          ToBoxes[Apply[bb, 
           Riffle[Map[bb[dd, #] &, Select[({vars}^{inds}), (# =!= 1 &)]],
             sp]]]]],
     Derivative /: 
      MakeBoxes[Derivative[inds__][g_][vars__Symbol], 
       form : TraditionalForm | DAFX] :=
      Module[{bb, dd, sp, vd},
        MakeBoxes[dd, _] ^= 
           If[Length[{inds}] == 1, "\[DifferentialD]", "\[PartialD]"];
        MakeBoxes[sp, _] ^= "\[ThinSpace]";
        vd[f_, v__, fmt_] := DisplayForm@ToBoxes[f[v], fmt];
        bb /: MakeBoxes[bb[x__], _] := RowBox[Map[ToBoxes[#] &, {x}]];
        FractionBox[ToBoxes[bb[dd^Plus[inds], vd[g, vars, form]]], 
          ToBoxes[Apply[bb, 
           Riffle[Map[bb[dd, #] &, Select[({vars}^{inds}), (# =!= 1 &)]],
              sp]]]]]
   ]
  ];
  Message[AbleitungsForm::ON,
    OptionValue[AuchStandard], OptionValue[MitArgumenten]];)

AbleitungsForm::noset="AbleitungsForm ist nicht aktiv.";  
AbleitungsForm[Off] :=
 (If[Position[FormatValues[Derivative],DAFX]!={},
    abFOpts={};
    (FormatValues[Derivative] =
       Delete[FormatValues[Derivative], 
           Position[FormatValues[Derivative], DAFX][[1, 1]]];);
    Message[AbleitungsForm::OFF], 
    Message[AbleitungsForm::noset]];)

AbleitungsForm[]:=
  If[abFOpts==={},
   Message[AbleitungsForm::noset],
   Message[AbleitungsForm::ON,abFOpts[[1]],abFOpts[[2]]]]; 

You may switch the display of derivatives on or off by calling AbleitungsForm[On] or AbleitungsForm[Off].
AbleitungsForm[] yelds a message displaying the status

Normally this changes the display for TraditionalForm only. There are two Options: Option AuchStandard->True makes it work in StandardForm too Option Mit Argumenten->True schows the functions Arguments.

Some Remarks

The function changes the FormatValues of Derivative. To see, where in the FormatValues these changes took place, I added the meaningless DAFX to the form parameter, which is used when switching Off.

As @Sjoerd pointed out, you can't paste the displayed expression as is (it's a Form). But this works:

equ = f''[x] == x f'[x]

equ // FullForm
DSolve[equ, f[x], x]

![displayed]: http://i.stack.imgur.com/SNEOs.png

Answer 4

What internally makes the superscript behave as a Derivative seems to be implemented with TagBox, this is what the output looks like:

SuperscriptBox["f", 
TagBox[
  RowBox[{"(", 
     RowBox[{"0", ",", "1"}], ")"}],
  Derivative],
MultilineFunction -> None]

If you show this with DisplayForm you will get something that looks like a superscript but evaluates as a Derivative. There are some similar examples mentioned in the documentation of TagBox. There seems to be no way to give this as input (except for inserting the TagBox by hand in the raw cell expression, of course...).

Edit: Of course one should never say it can't be done. I just stumbled over InputAliases in a post to mathgroup, a feature I usually don't use and thus always forget. Of course that would let you define a custom shortcut to insert exactly that box-expression, e.g. by evaluating the following:

CurrentValue[EvaluationNotebook[], InputAliases] = 
 Append[CurrentValue[EvaluationNotebook[], InputAliases], 
  "drv" -> SuperscriptBox["\[Placeholder]", 
    TagBox[RowBox[{"(", RowBox[{"\[Placeholder]"}], ")"}], 
     Derivative], MultilineFunction -> None]
  ]

then use EscdrvEsc to insert a corresponding template which you can fill by jumping from placeholder to placeholder using Tab. You could of course alternatively add such rules to your preferred stylesheet.

Answer 5

There is another way to input derivatives for people who really like TraditionalForm. I like TraditionalForm output to look as much as possible like a $\LaTeX$ typeset formula, and that's why I came up with the format that Peter is using in his answer. For example, I want partial derivatives to look like this: $\frac{\partial^2 f}{\partial x\partial y}$ or like this: $\frac{\partial^2 f(x,y)}{\partial x\partial y}$

The first one is achieved with the code I posted in this MathGroup thread.

But the more we tweak the output to look like classical math typesetting, the more incongruous the InputForm and StandardForm will look compared to the output.

I think the question in this post really is concerned with this disconnect. So I thought it's worth addressing how this can be bridged if you do decide to massage TraditionalForm into the above, more polished, form.

One possible way is to enter equations in TraditionalForm, too (not just output them that way). With the default settings, Mathematica doesn't exactly make this completely smooth (because it rightfully wants to avoid the potential ambiguities of TraditionalForm input). But it can be done.

To enter a partial derivative like the one above in the same form as above, the steps are as follows (trying to give a detailed description, but assuming you know how to input superscripts etc.):

1. Start a new input cell
2. Enter the name of the function f
3. Use keyboard or mouse to highlight the f and go to the menu item Cell > Convert to > TraditionalForm (or use the keyboard shortcut)
4. Now the f should have turned into an italic $f$ and you can continue by editing this cell:
5. Create a fraction (ctrl-/), add partial derivative symbols $\partial$ (escpdesc) exactly following the visual form of the example displayed above (including powers $\partial^2$ entered exactly like normal powers). For function arguments, use round parentheses $(x,y)$.

Now you can evaluate the cell. Mathematica will ask if you want to evaluate the input, and we have to confirm that we do. The point of this exercise is that you can in principle input expressions for (partial) derivatives in exactly the same form as they look in the $\LaTeX$-like TraditionalForm output that you get with my modification or the one in Peter's answer.