Finding the Jacobian

[Calcu]

Problem solved

 

[galactus]

The given points create 4 lines that make up the boundary of the region.

\(\displaystyle x+y=4, \;\ x+y=1, \;\ \frac{-x}{2}+y=1, \;\ \frac{-x}{2}+y=\frac{5}{2}\)

So, we can let \(\displaystyle u=x+y, \;\ v=\frac{-x}{2}+y\)

Solve for x and y in terms of u and v:

\(\displaystyle x=\frac{2}{3}(u-v)\)

\(\displaystyle y=\frac{1}{3}u+\frac{2}{3}v\)

Now, we have \(\displaystyle u=1, \;\ u=4, \;\ v=1, \;\ v=\frac{5}{2}\)

Now, since \(\displaystyle T(u,v)=(x,y)\), find \(\displaystyle T(4,1), \;\ T(1,1), \;\ T(1,\frac{5}{2}), \;\ T(4,\frac{5}{2})\)

i.e. \(\displaystyle T(1,1)=\left(\frac{2}{3}(1-1), \;\ \frac{1}{3}(1)+\frac{2}{3}(1)\right)=(0,1)\)


The partials of x and y are \(\displaystyle \frac{{\partial}x}{{\partial}u}=2/3, \;\ \frac{{\partial}x}{{\partial}v}=-2/3\)

\(\displaystyle \frac{{\partial}y}{{\partial}u}=1/3, \;\ \frac{{\partial}y}{{\partial}v}=2/3\)

The determinant of the partials is \(\displaystyle 2/3\).

Is that all you were given?. Just the coordinates?.

 

[Calcu]

Thanks so much for your help. I think I got all of that figured out now. I just have to evaluate the integral now but I am going to see if I can do that myself if not I may be back for more help.

Thanks again.

 

[Calcu]

Thanks, problem solved