Change of Variables
Table of Contents
Sometimes, we want to perform a change of variables between one coordinate system to another. For example, if we have \(\text{d}A=\text{d}x\text{ d}y\), we want to change this system to \(\text{d}A'=\text{d}u\text{ d}v\). This type of transformation is often performed when it is easier to do calculations in another coordinate system.
The issue is, the area \(A\) in \(xy\) coordinates may not be the same area in \(uv\) coordinates. Our job, then, is to figure out the relationship between \(\text{d}A\) and \(\text{d}A'\). In other words, we want to find a scaling factor that relates these two coordinate systems.
1. General Case
In general, let's say we have a coordinate system in \(uv\) such that its relationship with the \(xy\) coordinate system is as follows:
\begin{aligned} u=u(x,y) \\ v=v(x,y) \end{aligned}Then, by the approximation formulas, we can write:
\begin{aligned} \Delta u \approx u_x \Delta x + u_y \Delta y \\ \Delta v \approx v_x \Delta x + v_y \Delta y \end{aligned}Writing in matrix form, we have:
\begin{aligned} \begin{bmatrix} \Delta u \\ \Delta v \end{bmatrix} \approx \begin{bmatrix} u_x & u_y \\ v_x & v_y \end{bmatrix} \begin{bmatrix} \Delta x \\ \Delta y \end{bmatrix} \end{aligned}We can think of this as if we have a small rectangle in the \(xy\) system, it will be transformed into a small parallelogram in the \(uv\) system. Our system of equations tell us that the area of this parallelogram, transformed from a unit square in \(xy\), is the determinant like so:
\begin{aligned} |J| = \left| \begin{matrix} u_x & u_y \\ v_x & v_y \end{matrix} \right| \end{aligned}We call this value \(J\) the Jacobian. Then,
\[ \text{d}u\text{ d}v = |J|\text{ d}x\text{ d}y \]