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[parent] transformation between cartesian coordinates and polar coordinates (Example)

From the definition of a contravariant vector (contravariant tensor of rank 1)

$\displaystyle \bar{T}^{i} = T^{j}\frac{\partial \bar{x}^{i}}{\partial x^{j}}$ (1)

we get the transformation matrix from the partial derivatives

$\displaystyle A_{ij} = \frac{\partial \bar{x}^{i}}{\partial x^{j}}$ (2)

In order to calculate the transformation matrix, we need the equations relating the two coordinates systems. For cartesian to polar, we have

$\displaystyle r = \sqrt{ x^2 + y^2 } $

$\displaystyle \theta = tan^{-1}\left( \frac{y}{x} \right) $

and for polar to cartesian

$\displaystyle x = r \cos \theta $

$\displaystyle y = r \sin \theta $

So if we designate $(x,y)$ as the bar coordinates, then the transformation components from polar coordinates $(r,\theta)$ to cartesian coordinates $(x,y)$ is calculted as

$\displaystyle A_{11} = \frac{\partial \bar{x}^{1}}{\partial x^{1}} = \frac{\partial x}{\partial r} = \cos \theta$

$\displaystyle A_{12} = \frac{\partial \bar{x}^{1}}{\partial x^{2}} = \frac{\partial x}{\partial \theta} = -r \sin \theta$

$\displaystyle A_{21} = \frac{\partial \bar{x}^{2}}{\partial x^{1}} = \frac{\partial y}{\partial r} = \sin \theta$

$\displaystyle A_{22} = \frac{\partial \bar{x}^{2}}{\partial x^{2}} = \frac{\partial y}{\partial \theta} = r \cos \theta$

The components from cartesian coordinates to polar coordinates transform the same way, but now the polar coordinates have the bar

$\displaystyle B_{11} = \frac{\partial \bar{x}^{1}}{\partial {x}^{1}} = \frac{\partial r}{\partial x} = \frac{x}{\sqrt{x^2 + y^2}}$

$\displaystyle B_{12} = \frac{\partial \bar{x}^{1}}{\partial {x}^{2}} = \frac{\partial r}{\partial y} = \frac{y}{\sqrt{x^2 + y^2}}$

$\displaystyle B_{21} = \frac{\partial \bar{x}^{2}}{\partial {x}^{1}} = \frac{\partial \theta}{\partial x} = -\frac{y}{x^2 + y^2}$

$\displaystyle B_{22} = \frac{\partial \bar{x}^{2}}{\partial {x}^{2}} = \frac{\partial \theta}{\partial y} = \frac{x}{x^2 + y^2}$

In summary, the components of contravariant vectors in cartesian coordinates and polar coordinates transform between each other according to

$\displaystyle \left[ \begin{array}{c} x \ y \end{array} \right] = \left[ \beg... ...ta \end{array} \right] \left[ \begin{array}{c} r \ \theta \end{array} \right]$

$\displaystyle \left[ \begin{array}{c} r \ \theta \end{array} \right] = \left[... ... + y^2} \end{array} \right] \left[ \begin{array}{c} x \ y \end{array} \right]$



"transformation between cartesian coordinates and polar coordinates" is owned by bloftin.

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Cross-references: systems, matrix, tensor, vector
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This is version 11 of transformation between cartesian coordinates and polar coordinates, born on 2007-07-02, modified 2007-07-04.
Object id is 254, canonical name is TransformationBetweenCartesianCoordinatesAndPolarCoordinates.
Accessed 6286 times total.

Classification:
Physics Classification: 02.40.Hw (Classical differential geometry)
 04.20.Cv (Fundamental problems and general formalism)
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