Change in variables

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Change in Variables Overview:
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
Jacobian of a Transformation
- Definition of Jacobian
- Determinant of a 2 x 2 matrix
- Deals with derivatives
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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Finding the Transformations
Suppose we have $\, R$ $R$ , where $\, R \,$ $R$ is the region bounded by $\, y = x + 2, y = -x$ $y = x + 2, y = - x$ , and $\, y = \frac{x}{2}$ $y = \frac{x}{2}$ . Use the transformation $\, x = \frac{1}{3}u - \frac{1}{3}v \,$ $x = \frac{1}{3} u - \frac{1}{3} v$ and $\, y = \frac{1}{3}u + \frac{1}{3}v \,$ $y = \frac{1}{3} u + \frac{1}{3} v$ to determine the new region.
Suppose we have $\, R$ $R$ , where $\, R \,$ $R$ is the region bounded by $\, y = \frac{1}{x}, y = \frac{2}{x}, x = 2, x = 4$ $y = \frac{1}{x}, y = \frac{2}{x}, x = 2, x = 4$ . Use the transformation $\, x = 2u \,$ $x = 2 u$ and $\, y = \frac{v}{u} \,$ $y = \frac{v}{u}$ to determine the new region.
Finding the Jacobian
Given that the transformations are $\, x = 2u + 4v^{2} \,$ $x = 2 u + 4 v^{2}$ and $\, y = u^{2} - 4v$ $y = u^{2} - 4 v$ , find the Jacobian.
Given that the transformations are $\, x = u^{3}v^{5} \,$ $x = u^{3} v^{5}$ and $\, y = \frac{u}{v}$ $y = \frac{u}{v}$ , find the Jacobian.
Changing the Variables & Integrating
Evaluate $\, \int\int_{R} x - ydA \,$ $\int \int_{R} x - y d A$ where $\,R \,$ $R$ is the region bounded by $\, \frac{x^{2}}{4} + \frac{y^{2}}{9} = 1 \,$ $\frac{x ^{2}}{4} + \frac{y ^{2}}{9} = 1$ using the transformation $\, x = 2v, \,$ $x = 2 v,$ and $\, y = 3v$ $y = 3 v$ .
Evaluate $\, \int\int_{R} 2xydA \,$ $\int \int_{R} 2 x y d A$ where $\,R \,$ $R$ is the region bounded by $\, xy = 2, xy = 4, x = 2, x = 4\,$ $x y = 2, x y = 4, x = 2, x = 4$ using the transformation $\, x = 2u, \,$ $x = 2 u,$ and $\, y = \frac{v}{u}$ $y = \frac{v}{u}$ .

Topic Notes

Notes:

Transformations

Recall that in Integral Calculus, we can change the variable $x$ to $u$ of an integral using u-substitution. In other words, we can change from

$\int f(x)dx \to \int f(u)du$

We would like to do something similar like this with double integrals.

Transformations is about changing from one variable to another. We will first start by transforming regions.

Jacobian of a Transformation

The Jacobian of a transformation $x=g(u,v)$ & $y=h(u,v)$ is the following:

It is the determinant of a 2 x 2 matrix.

Change of Variables for a Double Integral

Suppose we want to integrate the function $f(x,y)$ in the region $R$ under the transformation $x=g(u,v)$ & $y=h(u,v)$ . Then the integral will now become:

$\int \int_R f(x,y)dA = \int \int_S f(g(u,v), h(u,v)) \left| \frac{d(x,y)}{d(u,v)}\right| du dv$

Change in variables

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Change in variables

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