# Change in variables

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##### Intros
###### Lessons
1. Change in Variables Overview:
2. Transformations
• Transformation = change one variable to another
• Similar to u-substitution in integral calculus
• $xy$-coordinate $\to$ $uv$-coordinate
• An Example of Change in Variable of Regions
3. Jacobian of a Transformation
• Definition of Jacobian
• Determinant of a 2 x 2 matrix
• Deals with derivatives
4. Change of Variables for a Double Integral
• All $x$'s & $y$'s become $u$'s & $v$'s
• Extra term is absolute value of Jacobian
• An Example
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##### Examples
###### Lessons
1. Finding the Transformations
Suppose we have $\, R$, where $\, R \,$ is the region bounded by $\, y = x + 2, y = -x$, and $\, y = \frac{x}{2}$. Use the transformation $\, x = \frac{1}{3}u - \frac{1}{3}v \,$ and $\, y = \frac{1}{3}u + \frac{1}{3}v \,$ to determine the new region.
1. Suppose we have $\, R$, where $\, R \,$ is the region bounded by $\, y = \frac{1}{x}, y = \frac{2}{x}, x = 2, x = 4$. Use the transformation $\, x = 2u \,$ and $\, y = \frac{v}{u} \,$ to determine the new region.
1. Finding the Jacobian
Given that the transformations are $\, x = 2u + 4v^{2} \,$ and $\, y = u^{2} - 4v$, find the Jacobian.
1. Given that the transformations are $\, x = u^{3}v^{5} \,$ and $\, y = \frac{u}{v}$, find the Jacobian.
1. Changing the Variables & Integrating
Evaluate $\, \int\int_{R} x - ydA \,$ where $\,R \,$ is the region bounded by $\, \frac{x^{2}}{4} + \frac{y^{2}}{9} = 1 \,$ using the transformation $\, x = 2v, \,$ and $\, y = 3v$.
1. Evaluate $\, \int\int_{R} 2xydA \,$ where $\,R \,$ is the region bounded by $\, xy = 2, xy = 4, x = 2, x = 4\,$ using the transformation $\, x = 2u, \,$ and $\, y = \frac{v}{u}$.