Using a Jacobian matrix to make a two-variable transformation

Jacobian transformations for functions in two variables

In the past we’ve converted multivariable functions defined in terms of cartesian coordinates $x$ and $y$ into functions defined in terms of polar coordinates $r$ and $\theta$.

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Similarly, given a region defined in the $uv$-plane, we can use a Jacobian transformation to redefine it in the $xy$-plane, or vice versa.

Given two equations $x=f(u,v)$ and $y=g(u,v)$, the Jacobian is

$\frac{\partial{(x,y)}}{\partial{(u,v)}}=\left|\begin{matrix}\frac{\partial{x}}{\partial{u}}&\frac{\partial{x}}{\partial{v}}\\ \frac{\partial{y}}{\partial{u}}&\frac{\partial{y}}{\partial{v}}\end{matrix}\right|=\frac{\partial{x}}{\partial{u}}\cdot\frac{\partial{y}}{\partial{v}}-\frac{\partial{x}}{\partial{v}}\cdot\frac{\partial{y}}{\partial{u}}$

How to use a Jacobian matrix to make a change of variables from one set of two variables to a different set of two variables

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Let’s do another example where we use the Jacobian to make a change of variables

Example

Find the Jacobian of the transformation.

$x=uv$

$y=2u-v^2$

Our functions tell us that we have a $2\times2$ set-up, so we’ll use the formula

$\frac{\partial{(x,y)}}{\partial{(u,v)}}=\frac{\partial{x}}{\partial{u}}\cdot\frac{\partial{y}}{\partial{v}}-\frac{\partial{x}}{\partial{v}}\cdot\frac{\partial{y}}{\partial{u}}$

given a region defined in the uv-plane, we can use a Jacobian transformation to redefine it in the xy-plane, or vice versa.

We need to start by finding the partial derivatives of $x$ and $y$ with respect to both $u$ and $v$.

$\frac{\partial{x}}{\partial{u}}=v$

$\frac{\partial{x}}{\partial{v}}=u$

and

$\frac{\partial{y}}{\partial{u}}=2$

$\frac{\partial{y}}{\partial{v}}=-2v$

We’ll plug the partial derivatives into our formula and get

$\frac{\partial{(x,y)}}{\partial{(u,v)}}=(v)(-2v)-(u)(2)$

$\frac{\partial{(x,y)}}{\partial{(u,v)}}=-2v^2-2u$

This is the Jacobian of the transformation.

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