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When performing a multiple integral with a transformation, e.g. $T:x=g(u,v), y=h(u,v)$, the following identity is true:

$$\iint_R f(x,y)dA = \iint_S f(g(u,v), h(u,v)) |J(u,v)| dA$$

where $R=T(S)$ and $$J(u,v)= \frac{\partial (x,y)}{\partial (u,v)} = \left | \begin{matrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{matrix} \right | = \frac{\partial x}{\partial u} \frac{\partial y}{\partial v} - \frac{\partial x}{\partial v} \frac{\partial y}{\partial u}$$

Is there an intuitive explanation for why $|J(u,v)|$ must be placed in the right hand side of the integral identity in order for one to integrate over $S$?

wheelix
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  • Where would you want to put it? On the left hand side of the equal sign? You can, but then it's going to be $J(x,y)=\frac{\partial(u,v)}{\partial(x,y)}$ – Andrei Nov 23 '21 at 05:32
  • Related: https://math.stackexchange.com/questions/267267/intuitive-proof-of-multivariable-changing-of-variables-formula-jacobian-withou. – Hans Lundmark Nov 23 '21 at 07:08
  • @Andrei my question is not asking why it must be on the right hand side, rather why the factor must be equal to $J(u,v)$ – wheelix Nov 27 '21 at 02:56
  • For that, I think the wikipedia entry is a good start. – Andrei Nov 27 '21 at 03:02

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