Complex partial derivative of a complex quadric

Questions
1

Hello everyone,

I want to use Mathematica to calculate the complex partial derivatives of complex quadric, for example:

x_1\bar x_1+x_2\bar x_2 -x_3\bar x_3 -1=0

Question

  1. I write \varphi =x_1\bar x_1+x_2\bar x_2 -x_3\bar x_3 -1=0 as

    [CurlyPhi] =
    Subscript[x, 1]*Conjugate[Subscript[x, 1]] +
    Subscript[x, 2]*Conjugate[Subscript[x, 2]] -
    Subscript[x, 3]*Conjugate[Subscript[x, 3]] - 1

is it right?

Question 2. I want to calculate for example

\frac{\partial^2 \varphi}{\partial x_2 \partial \bar x_3}

I write

c=D[D[\[CurlyPhi],[Subscript[x, 2]],Conjugate[Subscript[x, 3]]]

is it right?
Thank you in advance.

2

Hello Stefano,

long time ago I learnt that it is better not to use variables with a “complicated” structure like Superscript and so on. So it seems to be better to use “simple” expressions.

You could change that like

\[CurlyPhi] = 
 Subscript[x, 1]*Conjugate[Subscript[x, 1]] + 
  Subscript[x, 2]*Conjugate[Subscript[x, 2]] - 
  Subscript[x, 3]*Conjugate[Subscript[x, 3]] - 1

Then

\[CurlyPhi] = \[CurlyPhi] /. {
   Subscript[x, j_] :> ToExpression["x" <> ToString[j]],
   Conjugate[Subscript[x, j_]] :>  ToExpression["x" <> ToString[j] <> "c"]
   }

Your derivative is synthactically wrong. You should write

c = D[\[CurlyPhi], x2, x3c]

I hope this helps.

3

The conjugation has no derivative in the complex sense.