# Jacobian transformation [duplicate]

I would like to calculate the Jacobian transformation of the measure $dx dy$ $z = e^{x+iy}, \bar{z}=e^{x-iy}$ which should give $-2i|z|^2$. I am not sure how to do this in Mathematica. Further, I need it to generalize it to more variables. So, how could I write this transformation in "1-step"?

• What's the definition of the Jacobian that you use? Jun 19 '17 at 19:05
• The standard one, the matrix $h_{ij} = \partial f_{i}/\partial u^j$. Jun 20 '17 at 9:51
• Funny. Neither h, f or u appear in your question. Also, this is a matrix and in the question you said it should be a scalar. It would help us to help you if you gave a more detailed explanation of what you are looking for, or at least a derivation of the expected result (I got a different answer) Jun 20 '17 at 12:32
• Related, possible duplicate: How to make Jacobian automatically in Mathematica
– Jens
Jun 22 '17 at 4:26

z = Exp[x + I y];

zconj = Exp[x - I y];

vars = {x, y};

funcs = {z, zconj};

matJacobi = Outer[D, funcs, vars]

{{E^(x + I y), I E^(x + I y)}, {E^(x - I y), -I E^(x - I y)}}

detJacobi = Det[matJacobi]

-2 I E^(2 x)


but I do not get -2i Abs[z]^2, but -2i Exp[2x]

You can just use D. The Jacobian is:

j = D[{Exp[x + I y], Exp[x - I y]}, {{x, y}}]


and the Determinant of this is:

Det[j]

-2 I E^(2 x)