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Section 4-8 : 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