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$$\iint\limits_D (x − y)^2 \sin^2(x + y) \, dx \, dy$$ where $D$ is a parallelogram with vertices at $(π, 0), (2π, π), (π, 2π)$ and $(0, π)$.

We can change the variables as $$x=\frac{u-v}{2} \text{ and } y=\frac{u+v}{2}.$$ We can also do $$x=\frac{u+v}{2} \text{ and } y=\frac{u-v}{2}.$$ In both cases the limits of $u$ and $v$ are the same but the jacobian is the negative of one another (row is switched making it negative), thus integral turns out to be negative.

This is very confusing for me, because I am not able to understand why. Thanks for your help.

Note: In a few youtube videos I have seen that the absolute value of Jacobian is to be used, in the case no problem arises, but in the university I am studying in they have not specified to use the absolute value.

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They have most likely forgotten. Every change of variables requires the absolute value of the Jacobian.

This is because the sign the Jacobian has signifies the orientation given by the mapping. If the sign is negative then you reverse the orientation of the system originally considered, whereas if the sign is positive you maintain it.

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  • $\begingroup$ what should i do when jacobian depends on the variables itself, in this case it comes out constant , if it is of the form $f(u,v)$ then what should we do? $\endgroup$ – avz2611 Oct 6 '14 at 15:15
  • $\begingroup$ You take the absolute value as well. When the Jacobian varies with each point you should consider the regions on the considered domain where it is positive and negative. Consider as an example the Jacobian $J(x,y) = x-y$ and suppose the region is the unit square $0 \leq x \leq 1, 0 \leq y \leq 1$. You will consider the absolute value of $J(x,y)$ and check where $x-y \geq 0$ so $J(x,y) = x-y$ and where $x-y < 0$ so $J(x,y) = y-x$. $\endgroup$ – Mark Fantini Oct 6 '14 at 15:18
  • $\begingroup$ oh k, i did not know that , thanks for your help :) $\endgroup$ – avz2611 Oct 6 '14 at 15:19
  • $\begingroup$ @avz2611 Glad to help. Consider accepting it. $\endgroup$ – Mark Fantini Oct 6 '14 at 15:19

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