| Line 5: | Line 5: | ||
[[Category:math squad]] | [[Category:math squad]] | ||
| − | ==Jacobians== | + | == Jacobians and their applications == |
| − | + | by Joseph Ruan | |
| + | ---- | ||
===Basic Definition=== | ===Basic Definition=== | ||
| Line 50: | Line 51: | ||
---- | ---- | ||
<div style="font-family: Verdana, sans-serif; font-size: 14px; text-align: justify; width: 70%; margin: auto; border: 1px solid #aaa; padding: 2em;"> | <div style="font-family: Verdana, sans-serif; font-size: 14px; text-align: justify; width: 70%; margin: auto; border: 1px solid #aaa; padding: 2em;"> | ||
| + | </div> | ||
Revision as of 09:19, 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 and \frac{\partial x}{\partial v} = -u*\sin v y=u*\sin v \longrightarrow \frac{\partial y}{\partial u}= \sin v and \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
