# Change in variables

### Change in variables

#### Lessons

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:

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

• Introduction
Change in Variables Overview:
a)
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

b)
Jacobian of a Transformation
• Definition of Jacobian
• Determinant of a 2 x 2 matrix
• Deals with derivatives

c)
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