Hi, I have the following integral.(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\int\limits_{}^{} {\int\limits_R^{} {\left( {\sinh ^2 x + \cos ^2 y} \right)} \sinh 2x\sin 2ydxdy}

[/tex]

Where R is the part of the region 0 <= x, 0 <= y <= pi/2 bounded by the curves x = 0, y = 0, sinhxcosy = 1 and coshxsiny = 1.

In the hints section, there is a part which says [tex]J_{xy,uv} = \left( {\sinh ^2 x + \cos ^2 y} \right)^{ - 1} [/tex].

Firstly, to evaluate this integral I need to make a change of variables. The obvious ones are u = sinhxcosy and v = coshxsiny. Usually, to compute the Jacobian I would find expressions for x and y in terms of u and v. In this case this doesn't look possible.

The hint seems to have used [tex]\frac{{\partial \left( {x,y} \right)}}{{\partial \left( {u,v} \right)}} = \left[ {\frac{{\partial \left( {u,v} \right)}}{{\partial \left( {x,y} \right)}}} \right]^{ - 1} [/tex]. I know this is valid for some cases but I'm not sure which ones. Can someone explain to me when I can use the Jacobian relation given above?

Any help is appreciated.

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# Change of variables Jacobian

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