Module 1 : A Crash Course in Vectors
Lecture 2 : Coordinate Systems
  Example 3 :
  Show that the Jacobian of the transformation from cartesian to polar coordinates is $\rho$.
  Solution :
  We have
 
\begin{displaymath}J = \left\vert \begin{array}{cc} \frac{\partial x}{\partial\r... ...o } & \frac{\partial y}{\partial\theta} \end{array}\right\vert \end{displaymath}
  Using $x= \rho\cos\theta$and $y=\rho\sin\theta$, we have
 
\begin{displaymath}J = \left\vert \begin{array}{cc} \cos\theta & -\rho\sin\theta\\ \sin\theta & \rho\cos\theta \end{array}\right\vert = \rho \end{displaymath}

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