Module 1 : A Crash Course in Vectors
Lecture 2 : Coordinate Systems
  The Jacobian :
  When we transform from one coordinate system to another, the differential element also transform.
  For instance, in 2 dimension the element of an area is $dxdy$ but in polar coordinates the element is not $d\theta d\rho$ but $(\rho d\theta)d\rho$. This extra factor $\rho$ is important when we wish to integrate a function using a different coordinate system.
  If $f(x,y)$ is a function of $x, y$ we may express the function in polar coordinates and write it as $g(\rho,\theta)$. However, when we evaluate the integral $\int f(x,y)dxdy$ in polar coordinates, the corresponding integral is $\int g(r,\theta) \rho d\theta d\rho$. In general, if $ x= f(u,v)$ and $y = g(u,v)$, then, in going from $(x,y)$ to $(u,v)$, the differential element $dxdy \rightarrow \mid J\mid du dv$ where $J$ is given by the determinant
 
  The differentiations are partial, i.e., while differentiating $\partial x/\partial u = \partial f(u,v)/\partial u $, the variable $v$ is treated as constant. An useful fact is that the Jacobian of the inverse transformation is $1/J$ because the detrminant of the inverse of a matrix is equal to the inverse of the determinant of the original matrix.
  Example 3
   
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