I want to evaluate a matrix that has a function named alpha. When I take the derivative of alpha, I would like the result to give me an alpha'
Example: if I have sin(alpha) I want to get cos(alpha)alpha' but throughout the matrix.
It is quite unclear what you mean by stating that you have a "function" in Maple, of which you intend to take the derivative.
That could mean some expression depending upon a name such as t, with respect to which you intend on differentiating using Maple's diff command. And such an expression may be assigned to alpha, or it may contain the function call alpha(t).
Or perhaps you wish to treat alpha as an operator name, and differentiate functionally with Maple's D command.
I shall make a guess as to what you meant.
restart;
Typesetting:-Suppress(alpha(t));
Typesetting:-Settings(prime=t):
Typesetting:-Settings(typesetprime=true):
M := Matrix([[ sin(alpha(t)), exp(p*alpha(t)) ]]);
map(diff, M, t);
If that's not the kind of thing that you're after then you should explain your purpose and needs in greater detail.
Your question's title mentions a desire to have something not "evaluate". What did you mean by that? Have you perhaps assigned something to the name f?
Thank you for answering. I found the solution after a lot of guess and checking and reading. Here is my code with my solution.
with(Typesetting) :
Settings(typesetdot = true);
a:= alpha(t)
Rna:= [ cos(alpha), -sin(alpha), 0; sin(alpha), cos(alpha), 0; 0, 0, 1 ]
b := beta(t)
Rab:= [ cos(beta), -sin(beta), 0; sin(beta), cos(beta), 0; 0, 0, 1 ]
Rnab:= Rna . Rab
Rnab:= map(diff, Rnab, t)
Sorry for the multiple answers, I am getting use to the website.
Is there a way to get a list of variables used by a function?
For example:
a=1;
b=2;
f[x_]:= 2a*x+b;
Needed:
SomeFunction[f]
Output:
{{x},{a,b}}
The parameters of the function ({x}) are not really mandatory.
Thanks.
To get all the symbols (not necessarily variables) you could start with something like this:
DownValues[f]
which yields:
{HoldPattern[f[x_]] :> 2 a x + b}
The problem is then processing this in a way that you don't let Mathematica do the substitution. This is done with Hold:
held=(Hold //# DownValues[f][[1]])[[1, 2]]
which yields:
Hold[Hold[Hold[2] Hold[a] Hold[x]] + Hold[b]]
You can extract all the stuff that looks like a symbol with:
Cases[held, Hold[_Symbol], Infinity]
and you get:
{Hold[a], Hold[x], Hold[b]}
To make this a little nicer:
Union[Flatten[Hold ## Cases[held, Hold[_Symbol], Infinity]]]
which gives you:
Hold[a, b, x]
You still need the Hold, because as soon as you lose it Mathematica will evaluate a and b and you'll lose them as symbols.
If you notice, it's considering x a symbol, which you may not want because it's a parameter. You can tease the parameters out of the left side of the DownValues[f][[1]] RuleDelayed (:>) expression, and extract them, but I'll leave this detail to you.
I would like to know if it is possible to evaluate a table of functions at a point in mathematica.
Right now I am given a table of 10 functions and would like to evaluate them at x = 0.
I tried this :
Evaluate[myTable, {x, 0}]
And it does not evaluate anything, it just gives this output:
Sequence[ {term1, term2, term3, term4, term5, term6, term7, term8, term9, term10}, {x,0}]
Replacing term1 ... term10 with the actual terms.
How would I be able to do this?
Thanks,
Bucco
Through[{f1,f2,f3,f4,f5,f6,f7,f8,f9,10}[0]]
What is the best way to define a numerical constant in Mathematica?
For example, say I want g to be the approximate acceleration due to gravity on the surface of the Earth. I give it a numerical value (in m/s^2), tell Mathematica it's numeric, positive and a constant using
Unprotect[g];
ClearAll[g]
N[g] = 9.81;
NumericQ[g] ^= True;
Positive[g] ^= True;
SetAttributes[g, Constant];
Protect[g];
Then I can use it as a symbol in symbolic calculations that will automatically evaluate to 9.81 when numerical results are called for. For example 1.0 g evaluates to 9.81.
This does not seem as well tied into Mathematica as built in numerical constants. For example Pi > 0 will evaluate to True, but g > 0 will not. (I could add g > 0 to the global $Assumptions but even then I need a call to Simplify for it to take effect.)
Also, Positive[g] returns True, but Positive[g^2] does not evaluate - compare this with the equivalent statements using Pi.
So my question is, what else should I do to define a numerical constant? What other attributes/properties can be set? Is there an easier way to go about this? Etc...
I'd recommend using a zero-argument "function". That way it can be given both the NumericFunction attribute and a numeric evaluation rule. that latter is important for predicates such as Positive.
SetAttributes[gravUnit, NumericFunction]
N[gravUnit[], prec_: $MachinePrecision] := N[981/100, prec]
In[121]:= NumericQ[gravitUnit[]]
Out[121]= True
In[122]:= Positive[gravUnit[]^2 - 30]
Out[122]= True
Daniel Lichtblau
May be I am naive, but to my mind your definitions are a good start. Things like g > 0->True can be added via UpValues. For Positive[g^2] to return True, you probably have to overload Positive, because of the depth-1 limitation for UpValues. Generally, I think the exact set of auto-evaluated expressions involving a constant is a moving target, even for built-in constants. In other words, those extra built-in rules seem to be determined from convenience and frequent uses, on a case-by-case basis, rather than from the first principles. I would just add new rules as you go, whenever you feel that you need them. You probably can not expect your constants to be as well integrated in the system as built-ins, but I think you can get pretty close. You will probably have to overload a number of built-in functions on these symbols, but again, which ones those will be, will depend on what you need from your symbol.
EDIT
I was hesitating to include this, since the code below is a hack, but it may be useful in some circumstances. Here is the code:
Clear[evalFunction];
evalFunction[fun_Symbol, HoldComplete[sym_Symbol]] := False;
Clear[defineAutoNValue];
defineAutoNValue[s_Symbol] :=
Module[{inSUpValue},
s /: expr : f_[left___, s, right___] :=
Block[{inSUpValue = True},
With[{stack = Stack[_]},
If[
expr === Unevaluated[expr] &&
(evalFunction[f, HoldComplete[s]] ||
MemberQ[
stack,
HoldForm[(op_Symbol /; evalFunction[op, HoldComplete[s]])
[___, x_ /; ! FreeQ[Unevaluated[x], HoldPattern#expr], ___]],
Infinity
]
),
f[left, N[s], right],
(* else *)
expr
]]] /; ! TrueQ[inSUpValue]];
ClearAll[substituteNumeric];
SetAttributes[substituteNumeric, HoldFirst];
substituteNumeric[code_, rules : {(_Symbol :> {__Symbol}) ..}] :=
Internal`InheritedBlock[{evalFunction},
MapThread[
Map[Function[f, evalFunction[f, HoldComplete[#]] = True], #2] &,
Transpose[List ### rules]
];
code]
With this, you may enable a symbol to auto-substitute its numerical value in places where we indicate some some functions surrounding those function calls may benefit from it. Here is an example:
ClearAll[g, f];
SetAttributes[g, Constant];
N[g] = 9.81;
NumericQ[g] ^= True;
defineAutoNValue[g];
f[g] := "Do something with g";
Here we will try to compute some expressions involving g, first normally:
In[391]:= {f[g],g^2,g^2>0, 2 g, Positive[2 g+1],Positive[2g-a],g^2+a^2,g^2+a^2>0,g<0,g^2+a^2<0}
Out[391]= {Do something with g,g^2,g^2>0,2 g,Positive[1+2 g],
Positive[-a+2 g],a^2+g^2,a^2+g^2>0,g<0,a^2+g^2<0}
And now inside our wrapper (the second argument gives a list of rules, to indicate for which symbols which functions, when wrapped around the code containing those symbols, should lead to those symbols being replaced with their numerical values):
In[392]:=
substituteNumeric[{f[g],g^2,g^2>0, 2 g, Positive[2 g+1],Positive[2g-a],g^2+a^2,g^2+a^2>0,
g<0,g^2+a^2<0},
{g:>{Positive,Negative,Greater}}]
Out[392]= {Do something with g,g^2,True,2 g,True,Positive[19.62\[VeryThinSpace]-a],
a^2+g^2,96.2361\[VeryThinSpace]+a^2>0,g<0,a^2+g^2<0}
Since the above is a hack, I can not guarantee anything about it. It may be useful in some cases, but that must be decided on a case-by-case basis.
You may want to consider working with units rather than just constants. There are a few options available in Mathematica
Units
Automatic Units
Designer units
There are quite a few technical issues and subtleties about working with units. I found the backgrounder at Designer Units very useful. There are also some interesting discussions on MathGroup. (e.g. here).
I'd like to create a list of Hankel functions, defined in terms of an Nth derivative, but the Nth order derivatives get treated in the way that is described in the docs under "Derivatives of unknown functions", and left unevaluated. Here's an example:
Clear[x, gaussianExponential]
gaussianExponential[x_] := Exp[- x^2]
FullSimplify[Derivative[2][gaussianExponential[x]]]
I get:
(E^-x^2)^[Prime][Prime]
(instead of seeing the derivatives evaluated (and the final expressions are left unsimplified)).
Any idea what's going on here?
The correct syntax is:
Clear[x, gaussianExponential]
gaussianExponential[x_] := Exp[-x^2]
FullSimplify[Derivative[2][gaussianExponential][x]]
The Derivative applies to the function symbol f, not to the function evaluated at a point f[x]. So what you want is
Clear[x, gaussianExponential]
gaussianExponential[x_] := Exp[-x^2]
Derivative[2][gaussianExponential][x]//FullSimplify