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If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
Section 15.8 : Change of Variables
For problems 1 – 4 compute the Jacobian of each transformation.
- \(x = 4{u^2}v\hspace{0.25in}y = 6v - 7u\)
- \(x = \sqrt u \hspace{0.25in}\hspace{0.25in}y = 10u + v\)
- \(\displaystyle x = {v^3}u\hspace{0.25in}\hspace{0.25in}y = \frac{{{u^2}}}{v}\)
- \(x = {{\bf{e}}^u}\cos v\hspace{0.25in}y = {{\bf{e}}^u}\sin v\)
- 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 \(R\).
- 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 \(R\).
- 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 \(R\).
- 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 \(R\).
- 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 \(R\).
- 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 \(R\).
- 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 \(\displaystyle x = 2\sqrt {\frac{u}{v}} \), \(y = 4\sqrt {uv} \).
- 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 \(R\).
- 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 \(R\).
- 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 \(x = v\), \(\displaystyle y = \frac{{2u}}{{3v}}\).
- 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 \(R\).
- Derive a transformation that will transform the ellipse \(\displaystyle \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\) into a unit circle.
- Derive the transformation used in problem 12.
- Derive the transformation used in problem 13.
- 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.
- 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.