MHB Solving Change of Variables Problem with T(u,v) - Can You Help?

[Dizzy1]

Here's the problem:
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I'm having trouble using T(u,v) to figure out how to find the region that transforms into the the given region, R. Can anyone help?

Thank you!

 

[Prove It]

Have you at least drawn the region?

 

[Dizzy1]

Have you at least drawn the region?

yes

 

[Prove It]

Well now substitute $\displaystyle \begin{align*} x = u^{\frac{1}{3}}v^{\frac{2}{3}}\end{align*}$ and $\displaystyle \begin{align*} y = u^{\frac{2}{3}}v^{\frac{1}{3}} \end{align*}$ into all the functions. What are the new endpoints for u and v? What's the Jacobian?

 

[Dizzy1]

Well now substitute $\displaystyle \begin{align*} x = u^{\frac{1}{3}}v^{\frac{2}{3}}\end{align*}$ and $\displaystyle \begin{align*} y = u^{\frac{2}{3}}v^{\frac{1}{3}} \end{align*}$ into all the functions. What are the new endpoints for u and v? What's the Jacobian?

Don't I need to solve for u and v and plug in the points to figure out the region that transforms into R under T(u,v) ?