PlanetMath: polar coordinates

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polar coordinates

(Definition)

Let be Cartesian coordinates for $\mathbbmss{R}^2$ .

Then $r\ge 0$ , $\theta\in [0,2\pi)$ related to by

$\displaystyle x(r,\theta)$	$\displaystyle =$	$\displaystyle r \cos \theta,$
$\displaystyle y(r,\theta)$	$\displaystyle =$	$\displaystyle r \sin \theta,$

are the polar coordinates for

. It is simply written $(r,\theta)$ .

$\includegraphics{polar_4.eps}$

The polar coordinates of Cartesian coordinates $(x,y) \in \mathbbmss{R}^2\setminus\{0\}$ are

$\displaystyle r(x,y)$	$\displaystyle =$	$\displaystyle \sqrt{x^2+ y^2},$
$\displaystyle \theta(x,y)$	$\displaystyle =$	$\displaystyle \arctan (x,y),$

where $\arctan$ is defined here.

Polar basis. Polar coordinates are equipped with an orthonormal base $\{\mathbf{e_r,e_\theta}\}$ , which can be defined in terms of the standard cartesian base $\{\mathbf{i,j}\}$ in $\mathbbmss{R}^2$ as follows.

$\displaystyle \begin{bmatrix}\mathbf{e_r}\ \mathbf{e_\theta}\end{bmatrix}= \be... ...\sin\theta\mathbf{j}\ -\sin\theta\mathbf{i}+\cos\theta\mathbf{j}\end{bmatrix},$

where $\mathbf{e_r,e_\theta}$ are so-called radial and traverse or angular vector, respectively. Since these vectors are variable in direction, they are differentiable. In fact,

$\displaystyle \begin{bmatrix}\frac{d\mathbf{e_r}}{d\theta}\ \frac{d\mathbf{e_\... ...ix}= \begin{bmatrix}\hphantom{-}\mathbf{e_\theta}\ -\mathbf{e_r}\end{bmatrix}.$

The geometrical action of the derivative operator $d/d\theta$ is like a rotation operator that rotates each base vector a counter-clockwise angle equals to $\pi/2$ .

Position vector. For an arbitrary point of polar coordinates $(r,\theta)$ , its position vector comes given by the single equation

$\displaystyle \mathbf{r}=r\mathbf{e_r}.$

Relations with complex numbers. When the Euclidean plane $\mathbbmss{R}^2$ is identified with $\mathbbmss{C}$ by

$\displaystyle (x,y)\leftrightarrow x+yi,$

it is possible to define multiplications on $\mathbbmss{R}^2$ . Via polar coordinates, the formula for this multiplication becomes very simple, thanks to Euler's formula

$\displaystyle \cos{\theta}+i\sin{\theta}=e^{i\theta}.$

Thus, we have the following identification:

$\displaystyle (r,\theta)\leftrightarrow(x,y)\leftrightarrow x+yi= r\cos{\theta}+(r\sin{\theta})i=re^{i\theta}.$

If $P=(r_1,\theta_1)$ and $Q=(r_2,\theta_2)$ , the product of

and

works out to be $(r_1r_2,\theta_1+\theta_2)$ . (Even if one is not familiar with the complex exponential, this assertion may be checked directly using the angle sum identities for $\cos$ and $\sin$ .)

Multiplications of polar coordinates have some simple geometric interpretations. For example, if $R=(1,\alpha)$ and $Q=(r,\beta)$ , then $Q\rightarrow RQ$ given by $(1,\alpha)(r,\beta)=(r,\alpha+\beta)$ is the rotation of by angle $\alpha$ . If , then $(t,0)(r,\beta)=(tr,\beta)$ can be viewed as the scaling of along the ray $\overrightarrow{OQ}$ by . Note also that multiplication by has the same effect as multiplication by the scalar .

$\includegraphics{polar_5.eps}$

For more on polar coordinates, including their construction and extensions on domain of polar coordinates

and $\theta$ , see here.

Calculus in polar coordiantes. For reference, here are some formulae for computing integrals and derivatives in polar coordinates. The Jacobian for transforming from rectangular to polar cordinates is

$\displaystyle {\partial (x, y) \over \partial (r, \theta)} = r$

so we may compute the integral of a scalar field

$\displaystyle \int f (r, \theta) r dr d\theta .$

Partial derivative operators transform as follows:

$\displaystyle {\partial \over \partial x}$	$\displaystyle = \cos \theta {\partial \over \partial r} - {1 \over r} \sin \theta {\partial \over \partial \theta}$
$\displaystyle {\partial \over \partial y}$	$\displaystyle = \sin \theta {\partial \over \partial r} + {1 \over r} \cos \theta {\partial \over \partial \theta}$
$\displaystyle {\partial \over \partial r}$	$\displaystyle = \cos \theta {\partial \over \partial x} + \sin \theta {\partial \over \partial y}$
$\displaystyle {\partial \over \partial \theta}$	$\displaystyle = - r \sin \theta {\partial \over \partial x} + r \cos \theta {\partial \over \partial y}$

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"polar coordinates" is owned by CWoo. [ full author list (8) | owner history (1) ]

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Attachments:: construction of polar coordinates (Derivation) by CWoo

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Cross-references: Transform, partial derivative, field, Jacobian, integrals, Calculus, domain, extensions, scalar, ray, scaling, interpretations, angle sum identities, exponential, complex, even, product, simple, multiplications, Euclidean plane, complex numbers, equation, point, position vector, angle, rotates, rotation, operator, derivative, action, differentiable, variable, vector, radial, terms, base, orthonormal, basis, polar, Cartesian coordinates
There are 29 references to this entry.

This is version 11 of polar coordinates, born on 2005-04-24, modified 2008-01-20.
Object id is 6963, canonical name is PolarCoordinates.
Accessed 9329 times total.

Classification:

AMS MSC:

51-01 (Geometry :: Instructional exposition )

Pending Errata and Addenda

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