Change of Variables for double integral with exponential Using the substitution $u = x-y$ and $v = x+y$ , where $R$ is the is the first quadrant region bounded by the coordinate axes and the line $y = 1-x$ .
Evaluate $\iint_R e ^ { ( x-y) / ( x+y) } dx dy$ .
Any assistance will be appreciated.
 A: HINTS: 
I assume you know where to substitute in the $u$ and $v$. Consider what they equal, and that we perform this change of variables in order to simplify the whole expression! 
Now notice that the region $R$ is the triangular slice in the plane bounded by the lines $x=0,y=0,y=1-x$. What does this mean for the region $R^*$ in the $(u,v)$-plane (i.e., the region mapping between $R$)? Isolate $x,y$ and plug in the boundaries; see what you get! 
Using this information, you'd like your integral to be in the form $$\int_{R^*}e^{u/v}dudv=\int_{v=?}^{v=?}\int_{u=?}^{u=?}e^{u/v}dudv$$ where the limits of integration will depend on the information I talked about above. Hope that helps, and gives you an outline of the apporach in these types of problems.
A: Hint: $du/dx = 1$ so $du = dx$, and $dv/dy = 1$ so $dv = dy$. We also have $y = \frac{1}{2}(v-u)$ and $x = \frac{1}{2}(v+u)$. Find out what the preimage of $R$ (call it $R'$) under your coordinate transformation looks like, then directly integrate $\iint_{R'}e^{u/v}dudv$.
