Ilya Zakharevich on Fri, 29 Jan 1999 00:19:31 -0500

 Partial derivatives

```I'm thinking about adding new type "a transcendental function" to
PARI.  It is going to be a slight modification of a user variable
(undeterminate) with some additional rules of manipulation stored in
some database, so that

q=Trans(sin,x^2);
deriv(q,x)

may be calculated to be 2*x*Trans(cos,x^2).

But first I want to understand what happens with algebraic functions,
which *apparently* should be supported with the current implementation
as well.

Does not it look strange to you that

? q=Mod(z,x+y+z)
%1 = Mod(z, x + (z + y))
? deriv(q,x)
%2 = 0
? deriv(q,y)
%3 = 0
? deriv(q,z)
%4 = Mod(1, x + (z + y))

? Suppose that we have a modification

deriv(a,x,[y,y1,y2])

with the meaning "partial derivative of a wrt x when y,y1,y2 remain constant".

Then

deriv(q,x,[y])

might be calculated and be Mod(-1,x+(z+y)).

Any thoughts?

Ilya

P.S.  Why is it that I cannot make a damn from Chapter2's PolMods?  Is
not it a codification of some bugs in PARI?  Why should
y+Mod(x,x^2+1) behave any differently than Mod(x+y,x^2+1)?
```