transformation between cartesian coordinates and polar coordinates
TransformationBetweenCartesianCoordinatesAndPolarCoordinates
2007-07-02 21:59:30
2007-07-02 22:09:05
Example
tensor
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Calculating the transformation matrix from a cartesian coordinate system to a polar coordinate system illustrates the differences between a contravariant tensor and a covariant tensor of rank one. From the definition of a contravariant vector
\begin{equation}
\bar{T}^{i} = T^{j}\frac{\partial \bar{x}^{i}}{\partial x^{j}}
\end{equation}
we get the transformation matrix from the partial derivatives
\begin{equation}
A_{ij} = \frac{\partial \bar{x}^{i}}{\partial x^{j}}
\end{equation}
From the definition of a covariant vector
\begin{equation}
\bar{T}_{i} = T_{j}\frac{\partial x^{j}}{\partial \bar{x}^{i}}
\end{equation}
the corresponding transformation matrix is
\begin{equation}
B_{ij} = \frac{\partial x^{j}}{\partial \bar{x}^{i}}
\end{equation}
In order to calculate the transformation matrix, we need the equations relating the two coordinates systems. For cartesian to polar, we have
$$ r = \sqrt{ x^2 + y^2 } $$
$$ \theta = tan^{-1}\left( \frac{y}{x} \right) $$
and for polar to cartesian
$$ x = r \cos \theta $$
$$ y = r \sin \theta $$
So if we designate $(x,y)$ as the bar coordinates, then the transformtion components from polar coordinates $(r,\theta)$ to cartesian coordinates $(x,y)$ is calculted as
$$ A_{11} = \frac{\partial \bar{x}^{1}}{\partial x^{1}} = \frac{\partial x}{\partial r}$$
more to come soon...