Employ change of variables $u=x-y$, $v=x+y$ to evaluate the 2-dimensional integral: $\int \int (x-y)e^{x^2-y^2} dxdy$. This represents region R. R is the region bounded by the lines $x+y=1$ and $x+y=3$ and the curves $x^2 -y^2 =1$ and $x^2 - y^2 = -1$
I understand that there are 3 steps to changing the coordinates:
The first step, I substituted the new variable into the integrad :
$(x-y) e^{x^2 - y^2} = ue^{uv}$
Next, I need to know how to find the Jacobian:
$J(x,y,u,v) = \frac{\partial (x,y)}{\partial (u,v)}$
I have a problem with identifying what is the expression of $x$ and $y$ in order to do the partial differentiation for $u$ and $v$,
Next, is to find the new integration limits for $u$ and $v$, how do I go about finding that with this complicated region without a graphical calculator?
I've graphed it out on a graphical calculator and both $x$ and $y$ falls is in the region $ 0 \le x/y \le 1.667$
To find the new region from the transformation we'll need to apply the change of variables to whatever the region $R$ is in your question. For example, for some sort of triangle we might translate the vertices of that shape. I don't think I can help you more without seeing what the region $R$ referred to is.
To identify the expressions for your partial differentiation, try rearranging your two transformation equations and perhaps use elimination to find an equation in the form you want. For example: $$u=x-y, v=x+y$$ $$\implies{u+v=2x}$$ $$\implies{x=\frac{u+v}{2}}$$
$$\frac{\partial{x}}{\partial{u}}=\frac{1}{2}$$
Hopefully this helps finding the jacobian!