If I have two functions like these f[a_,b_] and g[a_,b_]. Lets say
f[a_,b_]:= a^2 + b
g[a_,b_]:= a + b
some times I encounter
Dt[f[a,b] , g[a,b]] and I want to have a defined value for this derivative,
like
Dt[f[a,b] , g[a,b]]-> R1
How can I define new derivative for this problem in Mathematica?
I mean whenever I differentiate f with respect g I get R1.
protected = Unprotect[Dt]
{"Dt"}
Dt[f[___], g[___]] := R1
Dt[f[a, b], g[]]
R1
Clear your definition
Clear@Dt
Dt[f[], g[]]
0
Remember
If your definition is no longer needed Clear it - otherwise you might get unwanted results in other areas. And restore protection:
Protect[Evaluate[protected]]
{"Dt"}
Problem with defining such derivatives is that Dt doesn't hold its arguments, so if f and g have some definitions
ClearAll[f, g]
f[a_, b_] := a^2 + b
g[a_, b_] := a + b
then they are evaluated when passed to Dt, so "standard trick" with defining UpValues like:
f /: Dt[HoldPattern@f[a_, b_], HoldPattern@g[a_, b_]] := r1[a, b]
will not work.
What you can do is to use an "environment", in which this derivative will have desired value.
ClearAll[f, g]
f[a_, b_] := a^2 + b
g[a_, b_] := a + b
ClearAll[withMyDerivative]
SetAttributes[withMyDerivative, HoldFirst]
withMyDerivative[expr_] :=
Block[{g},
g /: Dt[f[a_, b_], g[a_, b_]] := r1[a, b];
expr
]
Now, inside withMyDerivative environment, Dt[f[a_, b_], g[a_, b_]] will evaluate to r1[a, b].
g[a, b] (f[a, b] + Dt[f[a, b], g[a, b]]) // withMyDerivative
(* (a + b) (a^2 + b + r1[a, b]) *)
