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???.
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}}???
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.