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### Section 15.8 : Change of Variables

For problems 1 – 3 compute the Jacobian of each transformation.

1. $$x = 4u - 3{v^2}\hspace{0.25in}y = {u^2} - 6v$$ Solution
2. $$x = {u^2}{v^3}\hspace{0.25in}y = 4 - 2\sqrt u$$ Solution
3. $$\displaystyle x = \frac{v}{u}\hspace{0.25in}\hspace{0.25in}y = {u^2} - 4{v^2}$$ Solution
4. If $$R$$ is the region inside $$\displaystyle \frac{{{x^2}}}{4} + \frac{{{y^2}}}{{36}} = 1$$ determine the region we would get applying the transformation $$x = 2u$$, $$y = 6v$$ to $$R$$. Solution
5. If $$R$$ is the parallelogram with vertices $$\left( {1,0} \right)$$, $$\left( {4,3} \right)$$, $$\left( {1,6} \right)$$ and $$\left( { - 2,3} \right)$$ determine the region we would get applying the transformation $$\displaystyle x = \frac{1}{2}\left( {v - u} \right)$$, $$\displaystyle y = \frac{1}{2}\left( {v + u} \right)$$ to $$R$$. Solution
6. If $$R$$ is the region bounded by $$xy = 1$$, $$xy = 3$$, $$y = 2$$ and $$y = 6$$ determine the region we would get applying the transformation $$\displaystyle x = \frac{v}{{6u}}$$, $$y = 2u$$ to $$R$$. Solution
7. Evaluate $$\displaystyle \iint\limits_{R}{{x{y^3}\,dA}}$$ where $$R$$ is the region bounded by $$xy = 1$$, $$xy = 3$$, $$y = 2$$ and $$y = 6$$ using the transformation $$\displaystyle x = \frac{v}{{6u}}$$, $$y = 2u$$. Solution
8. Evaluate $$\displaystyle \iint\limits_{R}{{6x - 3y\,dA}}$$ where $$R$$ is the parallelogram with vertices $$\left( {2,0} \right)$$, $$\left( {5,3} \right)$$, $$\left( {6,7} \right)$$ and $$\left( {3,4} \right)$$ using the transformation $$\displaystyle x = \frac{1}{3}\left( {v - u} \right)$$, $$\displaystyle y = \frac{1}{3}\left( {4v - u} \right)$$ to $$R$$. Solution
9. Evaluate $$\displaystyle \iint\limits_{R}{{x + 2y\,dA}}$$ where $$R$$ is the triangle with vertices $$\left( {0,3} \right)$$, $$\left( {4,1} \right)$$ and $$\left( {2,6} \right)$$ using the transformation $$\displaystyle x = \frac{1}{2}\left( {u - v} \right)$$, $$\displaystyle y = \frac{1}{4}\left( {3u + v + 12} \right)$$ to $$R$$. Solution
10. Derive the transformation used in problem 8. Solution
11. Derive a transformation that will convert the triangle with vertices $$\left( {1,0} \right)$$, $$\left( {6,0} \right)$$ and $$\left( {3,8} \right)$$ into a right triangle with the right angle occurring at the origin of the $$uv$$ system. Solution