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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.