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"?
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2$\begingroup$ What's the definition of the Jacobian that you use? $\endgroup$– yohbsCommented Jun 19, 2017 at 19:05
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$\begingroup$ The standard one, the matrix $h_{ij} = \partial f_{i}/\partial u^j$. $\endgroup$– GorbzCommented Jun 20, 2017 at 9:51
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1$\begingroup$ 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) $\endgroup$– yohbsCommented Jun 20, 2017 at 12:32
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$\begingroup$ Related, possible duplicate: How to make Jacobian automatically in Mathematica $\endgroup$– JensCommented Jun 22, 2017 at 4:26
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2 Answers
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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]
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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)
