| Line 14: | Line 14: | ||
The Jacobian Matrix is just a matrix that takes the partial derivatives of each element of a function (which is in the form of a vector. Let F be a function such that | The Jacobian Matrix is just a matrix that takes the partial derivatives of each element of a function (which is in the form of a vector. Let F be a function such that | ||
| − | <math>F(u,v)=<x,y> </math> | + | <font size=4><math>F(u,v)=<x,y> </math></font> |
then the Jacobian matrix of this function would look like this: | then the Jacobian matrix of this function would look like this: | ||
| Line 24: | Line 24: | ||
To help illustrate this, let's do an example: | To help illustrate this, let's do an example: | ||
| − | Example #1 | + | Example #1: Let's take the Transformation: |
| + | |||
| + | <font size=5><math>T(u,v) = <u*\cos v,r*\sin v> </math> </font>. | ||
| + | |||
| + | What would be the Jacobian Matrix of this Transformation? | ||
Solution: | Solution: | ||
| Line 30: | Line 34: | ||
<math> | <math> | ||
| − | x=u*\cos v \longrightarrow \frac{\partial x}{\partial u}= \cos v | + | x=u*\cos v \longrightarrow \frac{\partial x}{\partial u}= \cos v \frac{\partial x}{\partial v} = -u*\sin v |
| − | y=u*\sin v \longrightarrow \frac{\partial y}{\partial u}= \sin v | + | y=u*\sin v \longrightarrow \frac{\partial y}{\partial u}= \sin v \frac{\partial y}{\partial v} = u*\cos v |
</math> | </math> | ||
Revision as of 09:23, 8 May 2013
Jacobians and their applications
by Joseph Ruan
Basic Definition
The Jacobian Matrix is just a matrix that takes the partial derivatives of each element of a function (which is in the form of a vector. Let F be a function such that
$ F(u,v)=<x,y> $
then the Jacobian matrix of this function would look like this:
$ J(u,v)=\begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix} $
To help illustrate this, let's do an example:
Example #1: Let's take the Transformation:
$ T(u,v) = <u*\cos v,r*\sin v> $ .
What would be the Jacobian Matrix of this Transformation?
Solution: Note that $ x=u*\cos v \longrightarrow \frac{\partial x}{\partial u}= \cos v \frac{\partial x}{\partial v} = -u*\sin v y=u*\sin v \longrightarrow \frac{\partial y}{\partial u}= \sin v \frac{\partial y}{\partial v} = u*\cos v $
Therefore the Jacobian matrix is
$ \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix}= \begin{bmatrix} \cos v & -u*\sin v \\ \sin v & u*\cos v \end{bmatrix} $
Now after doing
