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### Section 4-8 : Change of Variables

For problems 1 – 4 compute the Jacobian of each transformation.

1. $$x = 4{u^2}v\hspace{0.25in}y = 6v - 7u$$
2. $$x = \sqrt u \hspace{0.25in}\hspace{0.25in}y = 10u + v$$
3. $$\displaystyle x = {v^3}u\hspace{0.25in}\hspace{0.25in}y = \frac{{{u^2}}}{v}$$
4. $$x = {{\bf{e}}^u}\cos v\hspace{0.25in}y = {{\bf{e}}^u}\sin v$$
5. If $$R$$ is the region inside $$\displaystyle \frac{{{x^2}}}{{25}} + 49{y^2} = 1$$ determine the region we would get applying the transformation $$x = 5u$$, $$\displaystyle y = \frac{1}{7}v$$ to $$R$$.
6. If $$R$$ is the triangle with vertices $$\left( {2,0} \right)$$, $$\left( {6,4} \right)$$ and $$\left( {1,4} \right)$$ determine the region we would get applying the transformation $$\displaystyle x = \frac{1}{5}\left( {u - v} \right)$$, $$\displaystyle y = \frac{1}{5}\left( {u + 4v} \right)$$ to $$R$$.
7. If $$R$$ is the parallelogram with vertices $$\left( {0,0} \right)$$, $$\left( {4,2} \right)$$, $$\left( {0,4} \right)$$ and $$\left( { - 4,2} \right)$$ determine the region we would get applying the transformation $$x = u - v$$, $$\displaystyle y = \frac{1}{2}\left( {u + v} \right)$$ to $$R$$.
8. If $$R$$ is the square defined by $$0 \le x \le 3$$ and $$0 \le y \le 3$$ determine the region we would get applying the transformation $$x = 3u$$, $$y = v\left( {2 + {u^2}} \right)$$ to $$R$$.
9. If $$R$$ is the parallelogram with vertices $$\left( {1,1} \right)$$, $$\left( {5,3} \right)$$, $$\left( {8,8} \right)$$ and $$\left( {4,6} \right)$$ determine the region we would get applying the transformation $$\displaystyle x = \frac{6}{7}\left( {u - v} \right)$$, $$\displaystyle y = \frac{1}{7}\left( {10u - 3v} \right)$$ to $$R$$.
10. If $$R$$ is the region bounded by $$xy = 4$$, $$xy = 10$$, $$y = x$$ and $$y = 6x$$ determine the region we would get applying the transformation $$\displaystyle x = 2\sqrt {\frac{u}{v}}$$, $$y = 4\sqrt {uv}$$ to $$R$$.
11. Evaluate $$\displaystyle \iint\limits_{R}{{{x^2}{y^4}\,dA}}$$ where $$R$$ is the region bounded by $$xy = 4$$, $$xy = 10$$, $$y = x$$ and $$y = 6x$$ using the transformation $$\displaystyle x = 2\sqrt {\frac{u}{v}}$$, $$y = 4\sqrt {uv}$$.
12. Evaluate $$\displaystyle \iint\limits_{R}{{1 - y\,dA}}$$ where $$R$$ is the triangle with vertices $$\left( {0,4} \right)$$, $$\left( {1,1} \right)$$ and $$\left( {2,5} \right)$$ using the transformation $$\displaystyle x = \frac{1}{7}\left( {7 + u - v} \right)$$, $$\displaystyle y = \frac{1}{7}\left( {7 + 4u + 3v} \right)$$ to $$R$$.
13. Evaluate $$\displaystyle \iint\limits_{R}{{121x\,dA}}$$ where $$R$$ is the parallelogram with vertices $$\left( {0,0} \right)$$, $$\left( {6,2} \right)$$, $$\left( {7,6} \right)$$ and $$\left( {1,4} \right)$$ using the transformation $$\displaystyle x = \frac{1}{{11}}\left( {v - 3u} \right)$$, $$\displaystyle y = \frac{1}{{11}}\left( {4v - u} \right)$$ to $$R$$.
14. Evaluate $$\displaystyle \iint\limits_{R}{{\frac{{15y}}{x}\,dA}}$$ where $$R$$ is the region bounded by $$xy = 2$$, $$xy = 6$$, $$y = 4$$ and $$y = 10$$ using the transformation $$x = v$$, $$\displaystyle y = \frac{{2u}}{{3v}}$$.
15. Evaluate $$\displaystyle \iint\limits_{R}{{2y - 8x\,dA}}$$ where $$R$$ is the parallelogram with vertices $$\left( {6,0} \right)$$, $$\left( {8,4} \right)$$, $$\left( {6,8} \right)$$ and $$\left( {4,4} \right)$$ using the transformation $$\displaystyle x = \frac{1}{4}\left( {u - v} \right)$$, $$\displaystyle y = \frac{1}{2}\left( {u + v} \right)$$ to $$R$$.
16. Derive a transformation that will transform the ellipse $$\displaystyle \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$$ into a unit circle.
17. Derive the transformation used in problem 12.
18. Derive the transformation used in problem 13.
19. Derive a transformation that will convert the parallelogram with vertices $$\left( {4,1} \right)$$, $$\left( {7,4} \right)$$, $$\left( {6,8} \right)$$ and $$\left( {3,5} \right)$$ into a rectangle in the $$uv$$ system.
20. Derive a transformation that will convert the parallelogram with vertices $$\left( {4,1} \right)$$, $$\left( {7,4} \right)$$, $$\left( {6,8} \right)$$ and $$\left( {3,5} \right)$$ into a rectangle with one corner occurring at the origin of the $$uv$$ system.